�}URVIVAL MODEL§
AND THEIR ESTIMATION
Third Edition
Dick London, FSA
ACTEX Publications
Winsted, Connecticut
To all my teachers,
but most of all OGG, B 2 and GC
Copyright © 1988, 1997
by ACTEX Publications, Inc.
No portion of this book may be repro­
duced in any form or by any means
without the prior written permission of
the copyright owner.
Requests for permission should be addressed to
ACTEX Publications
P.O. Box 974
Winsted, CT 06098
Manufactured in the United States of America
10 9 8 7 6 5 4 3 2 l
Cover design by MUF
Library of Congress Cataloging-in-Publication Data
London, Dick, 1943Survival models and their estimation I Dick London, -- 2nd ed.
p. cm.
Bibliography: p.
Includes index.
ISBN 0-56698-268-5 (pbk.)
l. Insurance--Statistical methods.
2. Mortality--Statistical methods.
3. Population forecasting--Statistical methods.
I. Title
HG8782 .L57 1988
88-22118
304.6'4'015195--dc I 9
CIP
ISBN: 0-56698-268-5
CONTENT§
Preface to the Second Edition xi
Preface to the Third Edition xv
PART I
THE NATURE AND PROPERTIES OF
SURVIVAL MODELS
Chapter 1: INTRODUCTION
3
1.1 What is a Survival Model 3
1.2 Actuarial Survival Models 4
1.2. l The Select Model 5
1.2.2 The Aggregate Model 5
1.3 Forms of the Survival Model 6
1.4 Estimation 7
1.5 Study Design 8
1.5.1 Cross-Sectional Studies 9
l.5.2 Longitudinal Studies 9
1.6 Summary and Preview 10
Chapter 2: THE MATHEMATICS OF SURVIVAL MODELS
2.1 Introduction 13
2.2 The Distribution of T 13
2.2.l The Survival Distribution Function 13
2.2.2 The Cumulative Distribution Function 13
2.2.3 The Probability Density Function 14
2.2.4 The Hazard Rate Function 15
2.2.5 The Moments of the Random Variable T 16
2.2.6 Actuarial Survival Models 17
2.3 Examples of Parametric Survival Models 18
2.3.l The Uniform Distribution 18
2.3.2 The Exponential Distribution 19
2.3.3 The Gompertz Distribution 20
13
Contents
iv
2.3 .4 The Makeham Distribution 21
2.3.5 The Weibull Distribution 21
2.3.6 Other Distributions 21
2.3.7 Summary of Parametric Models 22
2.4 Conditional Measures and Truncated Distributions 22
2.4.1 Conditional Probabilities and Densities 22
2.4.2 Lower Truncation of the Distribution of X 24
2.4.3 Upper and Lower Truncation of the Distribution of X 25
2.4.4 Moments of Truncated Distributions 27
2.4.5 The Central Rate 28
2.4.6 Use of Conditional Probabilities in Estimation 29
2.5 Transformations of Random Variables 29
2.6 Mean and Variance of Transformed Random Variables 33
2.'/ Summary 35
2.8 Exercises 36
Chapter 3: THE LIFE TABLE
3.1 Introduction 39
3.2 The Traditional Form of the Life Table 40
3.3 Other Functions Derived from lx 42
3.3. l The Force of Mortality 43
3.3.2 The Probability Density Function of X 44
3.3.3 Conditional Probabilities and Densities 46
3.3.4 The Central Rate 49
3.3 .5 The Concept of Exposure 50
3.3.6 Relationship between nqx and ,,mx 53
3.4 Summary of Concepts and Notation 56
3.5 Methods for Non-Integral Ages 58
3.5.1 Linear Form for fx+s 58
3.5.2 Exponential Form for fx+s 61
3.5.3 Hyperbolic Form for .ex+s 63
3 .5.4 Summary 66
3.6 Select Life Tables 67
3.7 Summary 70
3.8 Exercises 71
39
Contents
v
PART II
ESTIMATION OF SURVIVAL MODELS
FROM SAMPLE DATA
Chapter 4: TABULAR SURVIVAL MODELS
ESTIMATED FROM COMPLETE DATA SAMPLES
19
4.1 Introduction 79
4.2 Study Design 79
4.3 Exact Time of Death 81
4.3.l The Empirical Survival Distribution 81
4.3.2 Analysis of the Empirical Survival Distribution 84
4.4 Grouped Times of Death 85
4.4.l Notation 85
4.4.2 The Distributions of N, and D, 86
4.4.3 Estimation of S(t) and Mo 87
4.4.4 Estimation of q, and p1 87
4.4.5 Estimation of S(/) from {p;} 89
4.4.6 Estimation of the Hazard Function 90
4.5 Summary 91
4.6 Exercises 93
Chapter 5: TABULAR SURVIVAL MODELS ESTIMATED FROM
. INCOMPLETE DATA SAMPLES: STUDY DESIGN
91
5.1 Introduction 97
5.2 Interval Estimates 99
5.3 Single- and Double-Decrement Environments 103
5.3.1 Uniform Distribution Assumptions 107
5.3.2 Exponential Distribution Assumptions I 08
5.4 Summary 110
5.5 Exercises 111
Chapter 6: TABULAR SURVIVAL MODELS ESTIMATED FROM
INCOMPLETE DATA SAMPLES:
MOMENT PROCEDURES
113
6.1 Introduction 113
6.2 Moment Estimation in a Single-Decrement Environment ·113
6.2.1 The Basic Moment Relationship 114
6.2.2 Special Cases 114
.
VI
Contents
6.2.3 Exposure 116
6.2.4 Grouping 118
6.2.5 Properties of Moment Estimators 119
6.2.6 A Different Moment Approach for Special Case C 122
6.3 Moment Estimation in a Double-Decrement Environment 124
6.3.1 The Basic Moment Relationships 124 ·
6.3.2 Uniform Distribution Assumptions 125
6.3.3 Exponential Distribution Assumptions 127
6.3.4 Hoem's Approach to Moment Estimation in a Double­
Decrernent Environment 129
6.4 The Actuarial Approach to Moment Estimation 130
6.4.1 The Concept of Actuarial Exposure 131
6.4.2 Properties of the Actuarial Estimator 133
6.5 Estimation of S(x) 134
/\
6.5. l Expected Value of S(x) 134
/\
6.5.2 Variance of S(x) 135
6.6 Summary 138
6.7 Exercises 139
Chapter 7: TABULAR SURVIVAL MODELS ESTIMATED FROM
INCOMPLETE DATASAMPLES:
MAXIMUM LIKELIHOOD PROCEDURES
7.1 Introduction 145
7.2 Single-Decrement Environment, Special Ca�e A 146
7.2.1 Partial Data 146
7.2.2 Full Data 148
.. 7.2.3 Summary of Special Case A 150
7 .3 Single-Decrement Environment, General Case 151
7.3. I General Form for Full Data 151
7.3.2 Full Data, Exponential Distribution 152
7.3.3 FuUOata,Uniform Distribution l.52
7.3.4 Partial Data, Special Case C 155
7.3.5 Special Case C with Random Censoring 158
7.3.6 Summary of Single-Decrement MLE's 159
7.4 Double-Decrement Environment 160
7.4.1 General Fonn for Full Data 160
7.4.2. Full Data, Exponential Distribution for Mortality 161
7.4.3 Full Data, Uniform Distribution for Mortality 161
145
Contents
vii
7.4.4 Partial Data, Special Case A 162
7.4.5 Partial Data (Special Case A), Exponential Distributions 163
7.4.6 Partial Data (Special Case A), Uniform Distributions 163
7.4.7 Summary of Double-Decrement MLE's 164
7.5 Properties of Maximum Likelihood Estimators 165
7.5.l Bias 165
7.5.2 Variance 165
7.5.3 Consistency and Efficiency 167
7.6 The Product-Limit Estimator 167
7.6.l Estimation of qx 167
7.6.2 Properties of the Estimator 170
7.6.3 Estimation of S(t) 171
7.6.4 The Nelson-Aalen Estimator 172
7.7 Summary 174
7.8 Exercises 175
Chapter 8: ESTIMATION OF PARAMETRIC SURVIVAL MODELS 181
8.1 Introduction 181
8.2 Univariate Models, Complete Data 181
8.2.l Exact Times of Death 181
8.2.2 Grouped Times of Death 190
8.3 Univariate Models, Incomplete Data 192
8.3.1 Maximum Likelihood Approaches 192
8.3.2 Least�Squares Approaches 197
8.4 Hypothesis Testing of Parametric Models 200
8.4.1 Grouped Times of Death 200
8.4.2 Exc;ict Times of Death 202
8.5 Concomitant Variables in Parametric Models 206
8.5.1 Bivariate (Select) Models 207
8.5.2 General Multivariate Models 210
8.5.3 The Additive Model 211
8.5.4 The Multiplicative Model 214
8.6 Summary 216
8.7 Exercises 216
Contents
viii
PARTIII
APPLICATIONS AND EXTENSIONS
Chapter 9: TRADITIONAL ACTUARIAL APPLICATIONS
9. l
9.2
9.3
9.4
9.5
9.6
225
Introduction 225
Actual Ages 225
9.2.l Decimal Years 226
9.2.2 Exact Ages 227
9.2.3 Calculation of Exposure 229
9.2.4 Grouping 231
9.2.5 Actual Age Summary 232
InsuringAges 233
9.3.1 Valuation Year ofBirth 233
9.3.2 Anniversary-to-Anniversary Studies 234
9.3.3 Select Studies 237
9.3.4 Insuring Age Summary 238
Fiscal Ages 239
9.4. l Fiscal Year ofBirth 239
9.4.2 Observation Periods for Fiscal Age Studies 240
9.4.3 New Members and Withdrawals 240
9.4.4 Fiscal Age Summary 242
Summary 242
Exercises 243
Chapter 10: MULTI-STATE MODELS
247
10.l Introduction 247
l 0.2 Progression ()fa Disease 248
10.2.1 · Description ofthe Model 248
10.2.2 Properties ofthe Model 249
10.2.3 Available Data 254
10.2.4 Estimating Model Parameters from Sample Data 256
10.2.5 Analysis ofthe Estimators 258
10.3 Modeling Continuing Care Retirement Community Populations 258
l 0.3.1 Description ofthe Model 259
10.3.2 Assumptions and Properties ofthe Model 260
l 0.3 .3 Transition Probabilities 260
l O .3 .4 Stochastic Processes 262
10.4 Exercises 265
Contents
.
1x
267
Chapter 11: APPLICATIONS IN ECONOMICS AND FINANCE
11 .1 Introduction 267
11.2 Duration of Economic Events 267
11.2.1 Duration of Unemployment 267
11.2.2 Model Selection 269
11.2.3 Uses of Economic Theory 272
11.2.4 Non-Parametric Estimation 273
11.2.5 Parametric Estimation 277
11.2.6 Use of Concomitant Variables 280
11.3 Mortgage Loan Prepayments 281
11.3.1 Default and Prepayment Risks 281
11.3.2 Overview of Mortgage-Backed Securities 282
11.3.3 Survey of Mortgage Prepayment Models 283
11.4 Corporate Bond Default Rates 289
11.4.1 Historical Approaches to Measuring Default Rates 289
11.4.2 Altman's Bond Mortality Rate Concept 291
11 .5 Exercises 292
Chapter 12: ADDITIONAL MISCELLANEOUS APPLICATIONS
295
12. l Introduction 295
12.2 Parametric Alternatives to Exponential Models 295
12.2.1 Piecewjse-Exponential Models 295
12.2.2 General Composite Hazard Rate Models 297
12.2.3 Polynomial Hazard Rate Models 303
12.3 Non-Parametric Models 308
12.3. l Product-Limit Approach 308
· 12.3.2 Actuarial Approach 313
12.4 Exercises 315
Appendix A: PROPERTIES OF ESTIMATORS
A.1 Bias 319
· A.2 Mean Square Error and Variance 320
A.3 Consistency 321
A.4 Efficiency 322
A.5 Distribution 322
319
Contents
x
Appendix B: ASYMPTOTIC PROPERTIES OF
MAXIMUM LIKELIHOOD ESTIMATORS
323
B. l Conditions 323
B.2 Results 324
B.3 Derivation of Equation (7 .61) 324
Appendix C: DERIVATION OF EQUATION (8.49)
327
Appendix D: EQUIVALENCE OF
EQUATIONS (10.22a) and (10.22b)
331
ANSWERS TO THEEXERCISES
333
Chapter 2 333
Chapter 3 334
Chapter 4 335
Chapter 5 336
Chapter 6 337
Chapter 7 338 .
Chapter 8 341
Chapter 9 343
Chapter IO 348
Chapter 11 349
Chapter 12 350
BIBLIOGRAPHY
351
INDEX
357
PREFACE TO THE SECOND EDITION
Survival Models and Their Estimation is a general textbook describing the
properties and characteristics of survival models, and statistical procedures
for estimating such models from sample data. Although it is written primarily
for actuaries, it is also intended to be of interest to a broader mathematical
and statistical audience. Academically, the text is aimed at the fourth year
undergraduate or the first year graduate level.
Actuaries and other applied mathematicians work with models which
predict the survival pattern of humans or other entities (animate or inani­
mate), and frequently use these models as the basis for calculations of con­
siderable financial importance. Specifically, actuaries use such models to cal­
culate the financial values associated with individual life insurance policies,
pension plans, and income loss coverages. Demographers and other social
scientists use survival models for making predictive statements about the
future make-up of a population to which the model is deemed to apply.
This text is not primarily concerned with the uses of survival models,
but rather with the question of how such models are established. This exer­
cise is sometimes referred to as survival model · development or survival
model construction; in this· text, however, we prefer the more descriptive
phrase survival model estimation.
It cannot be noted too strongly that the "real" survival distribution
(or survival probabilities) which apply to a group of persons is unknown, and
probably will forever be so. What we, therefore, attempt to do is estimate
that distribution, based on the data of a sample and a chosen estimation pro­
cedure. It is vitally important that this be clearly understood. Since the name
of the game is estimation, there are no "right" answers. There are only sound
(or unsound) procedures.
Because the result of our exercise is an estimate of the theoretical,
underlying, operative survival distribution, based on the particular experi­
ence of a sample, we recognize that the estimate is a realization of a random
variable, called an estimator. In tum, this random variable has properties
such as expected value and variance, and these properties tell us something
about the quality of the estimator. Note that we do not judge the "accuracy"
of the resulting estimate, but rather the quality (or validity) of the procedure
which produced the estimate. Properties of estimator random variables are
defined in Appendix A. Readers who are not entirely familiar with these
xii
Preface to tlte Second Editio11
properties may wish to review Appendix A before studying the specific esti­
mators developed in the text.
Frequently the estimated survival model produced directly from a
study is not entirely suitable for practical use, and is, therefore, systemati­
cally revised before such use. The process of revising the initial estimates
into revised estimates is called graduation. This step in the development of
a useable estimated survival model is the topic of a companion text to this
one entitled Graduation: The Revision ofEstimates.
Survival Models and Their Estimation is said to be a general text in
that it treats survival model estimation from the viewpoint of several differ­
ent practitioners, including the actuary, the demographer, and the biostatisti­
cian, without attempting to be an exhaustive treatment of any one of these
traditions.
A more thorough treatment of the actuarial tradition, from a different
perspective, can be found in texts by Gershenson [321, Benjamin and Pollard
[11], and Batten [8]; demographic approaches are the main theme of the
works by Keyfitz and Beekman [46], Spiegelman {71], and Chiang (19]; the
medical, or biostatistical, tradition is more deeply pursued by Elandt-Johnson
and Johnson [25). Additional texts, which deal with the statistical analysis of
survival data at the graduate level, include those by Lawless [50], Lee [51 ],
Miller [56], and Kalbfleisch and Prentice [42].
How is an initial estimated survival model determined from sample
data? There are many approaches to this. A survival model estimation prob­
lem will generally have three basic components: (1) the form and nature of
the sample data (which might also be called the study design); (2) the
chosen estimation' procedure; and (3) any simplifying assumptions made
along the way. AU of these concepts will be further developed in this text.
The traditional actuarial approach; for example, is characterized by a cross­
sectional study design using the transactional data of an insurance company
or pension fund operation, a method-of-moments estimation procedure, and
the Balducci distribution assumption. In this text we will consider, as well,
oth�r study designs, primarily those encountered by the clinical statistician or
the reliability engineer. In addition, we will consider other estimation proce­
dures, especially the maximum likelihood and product-limit methods. Final­
ly, we will consider other simplifying assumptions, such as the uniform and
exponential distributions.
The text presumes a basic familiarity with probability and statistics,
including the topics of estimation and hypothesis testing. The application of
these ideas specifically to the estimation of survival .models is then devel­
oped throughout the text. An effort has been made to keep the mathematics
Preface to the Second Edition
xiii
and the pedagogy at a level which does not require a prior familiarity with
the topic. Whenever a choice between mathematical rigor and pedagogic
effectiveness appeared to be necessary, we opted for the latter. As a result,
the level of mathematical rigor in the text may be somewhat less than that
desired by the precise mathematician, but the increased clarity which results
from sacrificing some rigor will hopefully be welcomed by the student reader.
The first edition of this text, published in 1986, included the subject
matter contained in the first eight chapters of the new edition. Chapters 5
and 6 have been completely rewritten from the first edition, Chapter 7 has
been substantially revised, and three entirely new chapters have been added.
The first edition was adopted by the Society of Actuaries as a reference for
its examination program in 1987, and many valuable suggestions for im­
provement were contributed by students and educators.
Drafts of the material in both editions of the text were submitted to a
review team, whose many valuable comments are reflected in the final
version. The indispensable assistance of this group is hereby gratefully ack­
nowledged.
Warre11 R. Luckner, FSA, of the Society of Actuaries, coordinated
the efforts of the review team and made many valuable comments himself.
Stuart A. Klugman, F SA, Ph.D., of the University of Iowa, was
particularly adept at detecting mathematical errors in the drafts, and much of
the precision that the text has attained is due to his careful efforts.
Stanley Slater, ASA, of Metropolitan Life Insurance Company, did a
remarkable job of editing the. drafts for improvements in writing style and
clarity, especially for the benefit of the student reader.
Other members of the review team who made valuable contributions
to the final text include Robert L. Brown, FSA, and Frank G. Reynolds, FSA,
both of the University of Waterloo, Cecil J. Nesbitt, FSA, Ph.D., of the
University of Michigan, Geoffrey Crofts, FSA, of the University of Hartford,
and Robert Hupf, FSA, of United of Omaha Life Insurance Company.
Much of the research and writing time invested in this project was
supported by a grant from the Actuarial Education and Research Fund. The
author would like to express his appreciation to the directors of AERF. for
this support.
Special thanks and appreciation are expressed to Marilyn J. Baleshi­
ski of ACTEX Publications who did the electronic typesetting for the entire
text, through what must have appeared to be an endless series of revisions.
Despite the efforts of the review team and the author to attain
pedagogic clarity and mathematical accuracy, errors and imperfections are
xiv
Preface to the Second Edition
undoubtedly still present in the text. For this the author and the publisher
take full responsibility and sincerely apologize to the reader. We respect­
fully request that you report these errors to the author at ACTEX
Publications, P.O. Box 974, Winsted, CT 06098.
Winsted, Connecticut
June, 1988
Dick London, FSA
PREFACE TO THE THIRD EDITION
It has been nearly ten years since the publication of the second edition of
Survival Models and Their Estimation, and its adoption by the Society of
Actuaries as the principal references for its Course 160 examination. It ap­
pears that the text has proved satisfactory in that role from the point of view
of exam candidate and examiner alike.
The mathematics of survival models themselves, and how they
might be estimated from sample data, has been a fairly stable topic, so that a
revision of the theory presented in the first eight chapters of the textbook
does not seem to be required at this time. Consequently, the reader familiar
with the second edition will note various clarifications and improvements in
presentation in these chapters, but no substantive change in the overall con'"
tent. Why, then, is a new edition appearing at this time?
Beginning in 1994, a Society of Actuaries Board Task Force on
Education has been working toward a new model of actuarial education for
the twenty-first century. Among many other important principles, the Task
Force has established that actuarial education in the future should include
guidance for the application of standard actuarial techniques in disciplines
beyond the traditional actuarial areas of insurance and pensions. Since
Survival· Models and Their Estimation plays its small part in the actuarial
education arena, as the reference text for Course 160, it naturally follows
that a revision of the text, guided by the Task Force principle of broadened
application, is now appropriate. The result is the appearance of new Chapters
I0, 11, and 12, which present applications of the general theory of survival
models in such fields as epidemiology, facilities planning, economics, in­
vestments, reliability engineering, and others.
Two other issues have affected certain changes from the prior to the
current edition of this text as well.
The first is that Chapter 9 in the prior edition, which described the
demographer's process of estimating survival models from general popula­
tion data, has been deleted from the text. The material in the prior edition
was based on out-of-date studies, namely the Canadian census of 1981 and
the U.S. census of 1980, and is not included in the course of reading for the
Course 160 exam. Furthermore, the content of that chapter is also included,
in up-dated form, in the new (third) edition of Robert L. Brown's Introduc­
tion to the Mathematics of Demography [16], and interested readers are
directed to that reference.
xvi
Preface to lite Tltird Edition
The second important change reflected in the new edition of this
text is a recognition that the traditional theory and data processing mechan­
ics for large-scale actuarial studies, developed in the pre-computer age,
should no longer receive the emphasis that it has in the past. Accordingly,
the description of the theory of the traditional actuarial approach and a
critique of that theory, as presented in Section 6.4, has been appropriately
reduced. In addition, Chapter 11 of the prior edition, which dealt with the
now out-of-date practice of calculating actuarial exposure as a by-product of
life insurance liability valuation, has been deleted. On the other hand, after
considerable deliberation it was decided that the prior Chapter 10 should be
retained as Chapter 9 in the new edition. Although dated in some respects, a
description of the actual data processing mechanics involved in the actuarial
technique of estimating survival models from insurance company and pen­
sion fund data, was deemed to still be an important component of actuarial
education.
The author would like to acknowledge the contributions of several
colleagues to the new edition of the text.
A review of Chapter l O was provided by Bruce Leonard Jones,
FSA, FCIA, Ph.D., of the University of Western Ontario. Frank G. Bensics,
FSA, Ph.D., of the College of Insurance suggested much of the content of
Chapter 11, and reviewed the drafts of that chapter as well. An important
contribution to the development of Chapter 11 was also made by Matthew J.
Hassett, ASA, Ph.D., of Arizona State University. A similar role for Chapter
12 was played by Rohan J. Dalpatadu, ASA, Ph.D., of the University of
Nevada at Las Vegas. Overall guidance for the content of the new edition
was provided by Robert A. Conover, FSA, the Education Actuary for Course
160 at the Society of Actuaries.
The text layout design and typesetting was again handled by
Marilyn J. Baleshiski at ACTEX Publications. The author is very apprecia­
tive of her considerable skills, professionalism, and patience.
We hope that readers familiar with the prior edition of this text will
agree that the changes reflected in the new edition make a valuable contri­
bution to actuarial education as we enter the twenty-first century. As always,
corrections and suggestions for improvements are welcome.
Winsted, Connecticut
April, 1997
Dick London, FSA
§URV1[VAL MODEL§
AND THEIR ESTIMATION
Third Edition
PART I
THE NATURE AND PROPERTIES
OF SURVIVAL MODELS
The main topic of this text is the statistical estimation of survival models and
the analysis of those estimated models.
Before we tackle the estimation idea, however, we must first develop a con­
siderable familiarity with survival models themselves, and that is the purpose
of the first three chapters of the text.
Chapte r 1 introduce s the ge ne ral ide a of survival mode ls in a conceptual
manner, and gives an overview of the entire text.
Chapte r 2 pre se nts a symbolic analysis of the survival mode l, and gives
several e xample s of distributions that might be used as parametric survival
models.
Chapter 3 describes the nature and properties of the traditional tabular
survival model, the life table. A strong effort is made in this chapter to show
that life tables, assisted by mortality distribution assumptions, have the same
capabilities as the parametric models of Chapter 2.
CHAPTER I
INTRODUCTION
J.1 WHATISASURVIVALMODEL
A survival model is a probability distribution for a special kind of random
variable.
Suppose an air conditioning unit is being operated in a laboratory
with the room temperature kept at a very high level. The unit begins to func­
tion at time t = 0, and we are interested in the probability that the unit will
still be functioning at any future time t in general. Symbolically we will
denote this probability by S(t).
The random variable we are considering here, called T, is defined to
be the time of failure of the entity known to exist at time t = 0, and is there­
fore frequently called the failure time random variable. Now if T is the
time of failure, then the probability of still functioning at time t is the same
as the probability that the failure time is later (mathematically greater) than
the value oft. Formally,
S(t)
= Pr(T> t).
(l.l)
By the nature of Tit is clear that T � 0, that S(O) = I, and that S(t) is a non­
increasing function. We will assume lim S(t) = 0.
t-oo
Now if Tis the time of failure of an entity which exists at t = 0, then
T is also the future lifetime of this entity measured from t = 0. Writers on
this subject will somewhat freely interchange these equivalent definitions of
T as "failure time" or "length of future lifetime." Although some consistency
might be convenient, the reader should have no difficulty in dealing with
these interchanges.
It is important to note in the above example that we are interested in
the survival of the cooling unit as a function of time since an initial event,
which occurred at time t = 0. In this case, the initial event is the starting up
of the unit under the test conditions. In particular, it is important to note that
we are not concerned with the age of the unit at the time.
As a second example, consider a study of the survival of laboratory
animals injected with a carcinogenic substance. The injection constitutes the
initial event which defines time t = 0. We are then_ interested in observing
the survival pattern of these animals as a function of time since injection.
4
Introduction
In each of the above two examples, the random variable of interest is
T, the time to failure. The age of the study unit, animate or inanimate, at
time t = 0, and hence its attained age at time of failure, is not of interest to
us, and might very well not even be known. The reason for this is that we
believe the chance of failure to be a function of time under the study condi­
tions, and not of the attained chronological age of the study unit. For this
reason we use the notation S(t).
For the most part, cases where failure (or death) is reasonably
viewed as a function of time since the initial event, and not of the chrono­
logical age of the study unit, will involve mechanical objects or laboratory
animals, as in the above two examples. On the other hand, cases dealing
with the survival patterns of humans, especially those of direct interest to
actuaries, will normally recognize that chance of failure (death) is related to
attained age. These cases where attained chronological age should be recog­
nized are further explored in the next section.
We do recognize some situations, however, where the nature of the
study group is such that attained chronological age is of minor importance,
compared to that of time since the initial event. For example, consider a
study of the survival of persons who have been diagnosed as having a certain
disease, and who have begun treatment for such disease. If time t = 0 is the
date of diagnosis, we might again be interested in measuring the survival of
such persons as a function of time since t = 0 only, if we believe that the
health condition (and the treatment program) would affect survival (or
failure) to such a great extent that age is practically immaterial. In such a
case we would again use S(t) for our survival function. Note that the various
persons entering the study would be of many different chronological ages at
t = 0, the time of their respective initial events.
1.2 ACTUARIAL SURVIVAL MODELS
The three examples given in Section 1.1 illustrate cases which art! of interest
primarily to the reliability engineer, the clinical statistician, or the biostatis­
tician. We stressed the idea that, in such cases, the actual chronological age
of each individual under study is not of importance, and might very well not
even be known.
In contrast, actuarial survival models, primarily for use in operating
insurance or pensiof} schemes, must recognize this chronological age, since
we believe that survival is a function of age. Two versions of actuarial
survival models will be explored.
Actuarial Survival Models
5
J.2.1 The Select Model
Consider a survival model to be used for making insurance calculations with
respect to persons selected for insurance coverage at age x (assumed to be an
integer). We see that the issuance of insurance constitutes the initial event
defining time t = 0 as described in the prior section, and the model is to give
probabilities of still being alive at time tin general. For example, if we are
still thinking in terms of our S(t) function, then S( l 0) gives the probability of
surviving to time t = 10 (probably measured in years). But surely we would
agree that S(lO) should have a different value if x = 25 when t = 0 than if
x = 55 at that time. In other words, in such cases S(t) alone is insufficient
for our needs. We need an S(t) which also depends, in one way or another,
on the value of x when t = 0. We will use the symbol S(t;x) in such cases.
In this context, the age at selection, x, is called a concomitant vari­
able. We still view time since selection, t, as our primary variable of interest,
but must reflect x, somehow, in our survival model S(t;x).
The standard actuarial approach is to simply have a separate S(t) for
each value of x. Then the survival model for each selection age is viewed as
a function of t only, but the appropriate model for use in any given case
would depend on x.
Age at selection is not the only concomitant variable that is known
to have an influence on survival. Another important one is sex, and this con­
comitant variable would probably also be reflected by having separate sur­
vival models for males and females. Still another example would be smokers
vs. nonsmokers. · If a survival model has been designed to reflect all three of
these concomitant variables, as well as the primary variable of time since
selection, then we might use the symbol S(t;x,m,s) to denote the model appr­
opriate for use with male smokers who were age x at selection.
We will further discuss these survival models with concomitant vari­
ables later in the text. For now we are mainly concerned with introducing
the idea that a factor other than time since the initial event can affect survi­
val, and that age at the initial event (selection) is an obvious example of this.
1.2.2 The Aggregate Model
Next we consider the special case where the initial event defining time t = 0
is the actual birth of the person whose survival probabilites are given by S(t).
Then, using x for attained age in general, we see that x = 0 when t = 0, so
attained age and elapsed time will run exactly together. In this case, one
could use either x or t for the primary variable, with the usual convention
being to use x. Thus the survival probabilities will be given by a function
called S(x), x � 0, with S(O) = 1 and S(x) - 0 as x - oo.
6
Introduction
In this case, the random variable Xis clearly the age at death, or the
future lifetime random variable, just as T was the time at death ( or time at
failure), or the future lifetime random variable.
It should be clear that S(x) and S(t) are really identical functions (but
different from S(t;x)). Thus as we explore the mathematical properties of
either S(t) or S(x), we are really dealing with both of them at the same time.
However, we prefer to use both symbols in this text, and to have a clear
distinction between the use of x and the use oft.
To summarize, whenever the chronological age of the entities whose
survival probabilities are being studied is not an important factor, so that
survival is viewed only as a function of time since some initial event, we will
use S(t). In the special case of the survival pattern of human beings from
actual chronological birth, we will use S(x). Both S(t) and S(x) are functions
of one variable. In the case of a study of the survival pattern of persons aged
x at t = 0, and where attained age is believed to affect survival, we will use
S(t;x). This will be the least common of the three cases.
1.3 FORMSOFTHESURVIVALMODEL
Thus far we have only referred to the function S(t) (or S(x)) conceptually,
without being very specific about its form. Since we did refer to T as a uni­
variate random variable, the reader might reasonably expect that we have
specific distributions in mind, such as the exponential or the normal, and this
is frequently the case. Whenever the survival probabilities, S(t), are given by
a mathematical formula, we say that S(t) is in parametric form. This name
is chosen since values of S(t) will depend on one or more parameters, as well
as on t. For example, if Tis an exponentially distributed random variable,
with S(t) = e-.>..,, then ,\ is a parameter in the survival model. Several speci­
fic parametric models are reviewed in the next chapter.
Traditionally, actuarial survival models have not been in parametric
form. The shape of S(x), as suggested by empirical data, is too complex to
be adequately represented by simple one-parameter distributions (such as the
uniform or the exponential). Better representation is accomplished by the
two-parameter Gompertz or Weibull distributions, and even better represen­
tation by the three-parameter Makeham distribution. (All five of these
distributions are reviewed in Section 2.3.)
Although these distributions have had some use as parametric survi­
val models, the actuarial profession has tended to favor tabular models over
parametric ones. A tabular survival model is one in which numerical values
of S(x) are presented for certain selected values of x, most commonly the
int�gers. It is thus a table of numbers, from which it gets its name.
Forms of the Survival Model
7
It has been argued that this table of numbers could still be said to
represent a parametric survival model, wherein each and every one of the
numbers is a parameter. There is some logic to this argument, but, as a
matter of definition, we prefer to call such a model tabular, and reserve the
term parametric for the situation defined above.
Now if a tabular model presents S(x) values only for x = 0, 1, ... ,
then the model lacks the ability to answer questions involving any fractional
values of x. We might say that the survival model is not totally specified.
To overcome this deficiency, it is customary to make an assumption about
the form of S(x) between adjacent integers. When such assumption, called a
mortality distribution assumption, is superimposed on the tabular model,
then S(x) is defined for all x, x � 0, and any calculation that could be made
from a parametric model can now be made from the tabular-model-with­
distribution-assumption combination.
The actuarial tabular survival model has been around for many
years. It is alternatively referred to as a life table, or mortality table, depend­
ing, apparently, on whether one is an optimist or a pessimist. For the form of
S(x) between adjacent integers, there are three common assumptions: linear
(by far the most common), exponential, and hyperbolic.
Because the traditional tabular survival model is so important, we
will describe it, along with the three common assumptions for S(x) between
adjacent integers, at considerable length in Chapter 3.
In the special case of our select model S(t-,x), actuarial uses of this
model are nearly always in tabular form. Recently the development of para­
metric forms for S(t-,x) in the actuarial context has been explored by
Tenenbein and Vanderhoof [74]. Many others have explored this idea for
S(t;x), as well as the more general S(t;z), where z is a vector of any number
of concomitant variables, in the clinical setting. (See, for example, Chapter
13 of Elandt-Johnson and Johnson [25], where additional references are also
given.) We briefly deal with this issue of parametric models with con­
comitant variables in Chapter 8, and give an illustration of such a model in
practice in Chapter 11.
JA ESTIMATION
Thus far we have defined the survival model as a function called S(t) (or
S(x)) which gives probabilities of surviving to time t (or age x), either as a
mathematical function of the variable, or as a table of numbers along with an
assumption of form between adjacent integers. It is our intent to continue
8
Introduction
this exploration of the nature and properties of these two forms of the
survival model in the next two chapters, so that a clear understanding of
them can be established. We will then proceed to the main business of this
text, which is to establish specific models for use. Because we take the point
of view that there exists a conceptual, underlying, operative survival model,
denoted by S(t) (or S(x)), then the specific model which we establish is an
approximation to, or an estimate of, that "real," operative model. Thus we
speak of estimating S(t), and will generally denote this estimate by S(t).
Various approaches to estimating S(t) will be employed, depending
on the nature of the sample data and the design of the study, and various
distribution assumptions will be utilized as well.
It is important to note the meaning of "estimating" with respect to
the two basic forms of the survival model. For the tabular model, we usually
estimate the conditional probabilities of survival over short unit intervals
(usually one-year periods), and produce our estimate of S(t) from them. (The
meaning of "conditional interval probabilities" will be clarified in the next
two chapters.) Although there will be a few variations from this approach, it
will be our most common route to S(t). Estimation of tabular models is
pursued in Chapters 4 through 7.
By contrast, with respect to a parametric su rvival model, estimation
of the unknown parameters of the assumed functional form of S(t) produces
the estimated S(t). Considerably fewer quantities are estimated here than in
the tabular case if we estimate these parameters directly from the sample
data. Frequently parametric survival model estimation is accomplished by
first estimating the sequence of conditional interval probabilities, and then
"fitting" them to the chosen parametric form. Both of these approaches,
along with the topic of hypothesis testing of parametric models, are covered
in Chapter 8.
1.5 STUDY DESIGN
In this text we will deal with approaches to survival model estimation that
are followed by the clinical statistician, the actuary, and the economist. In
this section we give a brief description of the traditional design of the study
with which these various practitioners deal. Survival model estimation is
also of great interest to the demographer. Demographic approaches to
estimating life tables from general population data are not treated exten­
sively in this text. The interested reader is referred to Brown [ 16].
Study Design
9
For the most part, it can be said that the actuary and the demo­
grapher deal with large sample studies, whereas most clinical studies are
based-on smaller samples. It is also generally true that actuarial and demo­
graphic studies have a cross-sectional design, whereas many clinical studies
have a longitudinal design.
1.5.1 Cross-Sectional Studies
In this study design, we begin by defining the study group, an identifiable
group of persons whose survival pattern is to be studied (i.e., whose under­
lying, operative survival model is to be estimated). Examples of the study
group would be the entire population of a city, state or nation (demographic),
or the policyholders of a life insurance company or members of a pension
plan (actuarial).
Next an observation period is chosen. At the beginning of this
period there are many persons who are already members of the study group
and they come under observation at that time. Others might join the study
group during the observation period, and some could drop out of the study
group without dying. This in-and-out movement during the study period is
referred to as migration. By appropriately classifying and manipulating the
data, especially the observed deaths, according to a chosen estimation proce­
dure, age-specific death probabilities are estimated from the sample data.
These probabilities constitute a tentative tabular survival model.
A major advantage of this study design is that the estimates can be
based on the up-to-date experience of a group moving through time, where,
in general, the time until failure is fairly lengthy.
1.5.2 Longitudinal Studies
In contrast to the large sample cross-sectional studies, clinical studies are
frequently quite different in their design. Rather than selecting an interval of
time, and observing failure data within that observation interval, these stud­
ies select a study group and follow the experience of that group longitud­
inally into the future. This study design is feasible when the time until
failure is, irt general, fairly short.
In many cases, clinical studies will have what we call a cohort
complete design. In this study design, a cohort, ot initial group, of study
units are selected. These may be light bulbs, injected mice, or persons under­
going a certain medical treatment. In all cases, each study unit is selected at
time t = 0. Note that the respective t = 0 times for each study unit could be
10
Introduction
the same date on the calendar (as would likely be true for the light bulb or
injected mice studies), or could be different dates (as would likely be true for
humans). Even in the latter case, however, the study would normally be
conducted with respect to time since initial selection, regardless of the dates
on which t = 0 occurred. (This matter will be further clarified by example in
Chapter 4.)
Once the initial group is selected, it is kept under observation until
all have died (failed), and the time of each death is recorded. It is clear that
in order to perform this type of study, the observer must have the ability to
control the study group, in the sense that none of the study units may be
allowed to "disappear" without dying (failing). For this reason, such studies
are sometimes called controlled-data studies. By its nature, this study design
is not feasible for large-sample studies of healthy lives; for such studies the
cross-sectional design is used.
In some cases the cohort, longitudinal study may be terminated
before all study units fail in order to avoid an unduly long study period. In
such cases we have an incomplete data study, rather than a complete data
study. Two common approaches to terminating the study exist. If the study is
terminated at a fixed, pre-determined date, then we say the data have been
truncated. On the other hand, if a study is continued until a pre-determined
number of deaths has been observed, then we say the data have been cen­
sored. The major reason for distinguishing between truncation and censor­
ing lies in the subsequent analysis of the estimation results.
1.6 SUMMARY AND PREVIEW
In this introductory chapter we have set the stage for a study of the nature
and properties of survival models and approaches to estimating these
models, which is the main topic of the text. The o�jectives of this chapter
were to define the concept of a survival model, to describe the two basic
forms of such a model, and to present a brief orientation to the idea of esti­
mating the model.
In Chapter 2 we will pursue the mathematical properties of the
conceptual survival model, and examine several specific parametric models.
In Chapter 3 we will explore the tabular survival model (the life table).
Chapters 4 through 8 comprise Part II of the text, and are concerned
with the theory of estimating survival models from sample data.
Summary and Preview
11
Chapter 4 addresses this issue in the convenient setting of complete
data samples, and the next four chapters all allow for incomplete data
samples.
Chapter 5 is devoted to an analysis of study design to discover the
types of estimation problems that we will encounter.
Chapter 6 describ es moment estimation, including the traditional
actuarial approach.
Chapter 7 deals with maximum likelihood estimation, including the
product limit estimator.
Chapter 8 deals with the estimation of parametric survival models,
from both complete and incomplete data samples. It includes the topic of hy­
pothesis testing of such estimated models, and a brief discussion of allowing
for concomitant variables.
Chapters 9 through 12 comprise Part III of the text, and present
various applications of the theory of survival models and their estimation as
presented in the first two parts.
Chapter 9 describes approaches to processing the data in large­
sample actuarial studies, with particular reference to life insurance company
and pension fund data.
Chapter 10 describes multi-state models, such as models for progres­
sive stages of a disease. A brief introduction to the topic of stochastic
processes is also included.
Chapter 11 deals with applications of survival model theory in the
fields of economics and finance. In particular we will consider the distribu­
tions of such variables as the duration of economic events, the time until
default of corporate bonds, and the time until prepayment ofmortgage loans.
Finally, Chapter 12 presents some applications of survival model
theory in other fields, primarily that of reliability engineering.
CHAPTER2
THE MATHEMATICS OF SURVIVAL MODELS
2.1 INTRODUCTION
Before we begin our exploration of the topic of estimating a survival model,
we need to develop a complete understanding of the nature of survival mod­
els themselves.
Since a survival model is a special kind of probability distribution,
most of the material in this chapter will be familiar to those with a good
knowledge of probability. Furthermore, the survival model is discussed in
many standard textbooks on actuarial mathematics. (See, for example, Bow­
ers, et al. [12].)
2.2 THE DISTRIBUTION OFT
2.2.1 The Survival Distribution Function
In Chapter. 1 we chose to define and describe a survival model in terms of
the function S(t), which represents Pr(T> t), where T is the failure time
random variable. This function of the random variable T is called the Sur­
vival Distribution Function (SDF). We recall that it gives the probability that
failure (death) will occur after time t, which is the same as the probability
that the entity, known to exist at time t = 0, will survive to at least time t.
We also recall that S(O) = 1 and S(oo) = 0.
2.2.2 The C umulative Distribution Function
The Cumulative Distribution Function (CDF) of T is F(t). The CDF gives
the probability that the random variable will assume a value less than or
equal to t. That is,
F(t)
= Pr(Ts. t).
(2.1)
In the special case of our failure time random variable, F(t) gives the
probability that failure (death) will occur not later than time t. It should be
clear that
14
The Mathematics of Survival Models
1 - S(t),
F(t)
(2.2)
and that F(O) = 0 and F(oo) = 1.
In most probability textbooks, the CDF, F(t), is given greater
emphasis than is the SDF, S(t). But for our special kind of random variable,
S(t) will receive greater attention.
2.2.3 The Probability Density Function
For the special case of a continuous random variable, the Probability Density
Function (PDF),f(t), is defined as the derivative of F(t). Thus
f(t)
= Ji F(t) =
d S(t), t 2: 0.
- dt
(2.3)
Consequently, it is easy to see that
1
F(t) = 1 f(y) dy,
{2.4)
Joe
j(y)dy.
(2.5)
l f(y)dy
= I.
(2.6)
and
=
S{t)
Of course it must be true that
oc
Although we have given mathematical definitions of f(t), it will be
useful to describe f(t) more fully in the context of the failure time random
variable. Whereas F(t) and S(t) are probabilities which relate to certain time
intervals,f(t) relates to a point of time, and is not a probability, per se. We
prefer to rder to f(t) by its conventional description as ''probability density."
It is the density of failure.at time t, and is an instantaneous measure, as
opposed to an interval measure.
It is important to recognize that f(t) is the unconditional density of
failure at time t. By this we mean that it is the density of failure at time t
given only that the entity existed at t = 0. The significance of this point will
become clearer in the next subsection.
The Distribution of T
15
2.2.4 The Hazard Rate Function
We have just established that the PDF of T,f(t), is the unconditional density
of failure at time t. We now define a conditional density of failure at time t,
such density to be conditional on survival to time t. This conditional instan­
taneous measure of failure at time t, given survival to time t, will· be called
the hazard rate at time t, or the Hazard Rate Function (HRF) when viewed
as a function oft. It will be denoted by >.(t).
In general, if a conditional measure is multiplied by the probability
of obtaining the condition, then the corresponding unconditional measure
will result. Specifically,
(Conditional density of failure at time t, given survival to time t)
x (Probability of survival to time t)
= (Unconditional density of failure at time t).
Symbolically this states that
= f(t),
>.(t) . S(t) or
)
>.(/
M
= S(t) ·
(2.7)
(2.8)
Mathematically, Equations (2.8) and (2.3) define the HRF and the
PDF of the failure time random variable, and these mathematical definitions
are, of course, very important. However, it is equally important to have a
clear understanding of the descriptive meanings of >.(t) and f{t). They are
both instantaneous measures of the density of failure at time t; they differ
from each other in that >.(t) is conditional on survival to time t, whereasf(t)
is unconditional (i.e., given only existence at time t = 0).
In the actuarial context of human survival models, failure means
death, or mortality, and the hazard rate is normally called the force of
mortality. We will discuss the actuarial context further in this chapter and in
Chapter 3.
Some important mathematical consequences follow directly from
Equation (2.8). Since f(t) = - S(t), it follows that
*,
>.(t)
=
- dt
IL S(t)
S(t)
= - dtd lnS(t).
(2.9)
The Mathematics of Survival Models
16
Integrating, we have
11
or
= - lnS(t),
>.(y)dy
= exp [ - { >.(y)dy].
S(t)
,
(2.10)
(2.11)
The Cumulative Hazard Function (CHF) is defined to be
A(t)
=
so that
11
>.(y)dy
= - In S(t),
= e-A(t)_
S(t)
(2.12)
(2.13)
2.2.5 The 1lfoments of the Random Variable T
The first moment of a continuous random. variable defined on [O, oo) is given
by
=
E[T]
loc
t · f(t)dt,
(2.14)
if the integral exists, and otherwise the first moment is undefined. Integra­
tion by patts yields the alternative formula
E[T]
=
1= S(t)dt,
(2, 15)
a form which is frequently used to find the first moment of a failure time
random variable.
The second moment of Tis given by
(2.16)
if the integral exists, so the variance of T can be found from
Var(T)
= E[T 2 ] - {E[T]}2.
(2.17)
The Distribution of T
17
Specific expressions can be developed for the moments of T for specific
fonns of/(t). This will be pursued in the following section.
Another property of the future lifetime random variable that is of
interest is its median value. We recall that the median of a random variable
is the value for which there is a 50% chance that Twill exceed ( and thus also
not exceed) that value. Mathematically,y is the median of T if
Pr(T?:. y)
= Pr(T <_Sy) = ! ,
(2.18)
so thatS(y) = F(y) = 2I .
2.2.6 Actuarial Survival Models
Thus far in this section we have considered only the random variable T, and
have looked at various quantities related to that random variable and the
interrelationships among those quantities. Exactly the same quantities and
relationships exist for the actuarial survival model represented by the SDF
S(x), x ?:. 0.
Special symbols are used in the actuarial context for some of the
concepts defined in this section. The hazard rate, called the force of mortal­
ity, is denoted by µx , rather than ..\(x). Thus
µx
=
- �S(x)
S(x)
d ln S(x).
= - dx
(2.9a)
It is also customary to denote the first moment of Xby �o- Thus
�o
= E[X] =
100
x · f(x)dx.
(2.19)
Since �o is the unconditional expected value of X, given only alive at x ,=,; 0,
it is called the complete expectation of life at birth.
For the select model S(t;x), recall that t is a value of the random
variable T, and x is the age at which the person to whom S(t;x) refers was se­
lected. The expected value of T, E[T;x], gives the expected future lifetime
(or expectation of life) for a person selected at age x, and is denoted by �(x]­
The HRF is denoted by µ[xJ+t, and is given by
µ[x)+t =
-
- f, S(t;x)
S(t;x)
= - dtd lnS(t;x).
(2.9b)
The Mathematics of Survival Models
18
We recognize that the moments of X or T given above are all uncon­
ditional. Conditional moments, and other conditional measures, are defined
conceptually in Section 2.4, and the standard actuarial notation for them is
reviewed in Chapter 3.
2.3 EXAMPLES OF PARAMETRIC SURVIVAL MODELS
In this section we explore several non-negative continuous probability distri­
butions which are candidates for serving as survival models. In practice,
some distributions fit better than others to the empirical evidence of the
shape of a failure time distribution, so we will comment on each distribution
we present regarding its suitability as a survival model.
2.3.J The Uniform Distribution
The uniform distribution is a simple two-parameter distribution, with a con­
stant PDF. The parameters of the distribution are the limits of the interval
on the real number axis over which it is defined, and its PDF is the recipro­
cal of that interval length. Thus if the random variable is defined over the
interval [a, b], then/(t) = b�a for a :S t :S b, and/(t) = 0 elsewhere.
For the special case of the future lifetime random variable, a= 0.
Therefore, b is the length of the interval, as well as the greatest value oft for
which/(t) > 0. When the uniform distribution is used as a survival model,
the Greek w is frequently used for this parameter, so the distribution is
defined by
f(t)
= b, 0 :S t :S w.
(2.20)
The following properties of the uniform distribution easily follow, and
should be verified by the reader:
S(t)
1
= fo J(y)dy - j
(2.21)
w
= I - F(t) = l f(y)dy = w w- t
(2.22)
= w -l t
= JJ!}
S(t)
(2.23)
F(t)
>.(t)
19
Examples of Parametric Survival Models
E[T]
=
w
lo
t J(t)dt
Var(T) = E[T 2 ] - {E[T]} 2
2
(2.24)
= 1;
(2.25)
The uniform distribution, as a survival model, is not appropriate over
a broad range of time, at least as a model for human survival. It is of histor­
ical interest, however, to note that it was the first continuous probability dis­
tribution to be suggested for that purpose, in 1724, by Abraham de Moivre.
The major use of this distribution is over short ranges of time (or
age). We will explore this use of the uniform distribution quite thoroughly
in Section 3 .5. I .
2.3.2 The Exponential Distribution
This very popular one-parameter distribution is defined by its SDF to be
S(t)
= e->.t, t 2'.: 0, .X 2'.: 0.
(2.26)
It then follows that the PDF is
f(t)
= - 1, S(t) - >-.e->.
1
,
(2.27)
so that the HRF is
- )..
.X(t) - f(t) - S(t)
'
(2.28)
a constant. In the actuarial context, where the hazard rate is generally called
the force of mortality, the exponential distribution is referred to as the con­
stant force distribution.
� EXAMPLE 2.1 lj Show that, for the exponential distribution,
E[T]
and
= x1
Var(T) -
1.
)..2
(2.29)
(2.30)
The Mathematics of Survival Models
20
I SOLUTION I
parts produces
have
E [T] =
ft t -j(t)dt = ft t · >. e->.t dt.
ft e->.t dt, whence E(T] = - ! e->.i[
x
r t2 . ). e->.t dt
2·
>.2
lo
Then
Var(T) =
Integration by
I
>:· We also
;2 -
{±r =
12 .
D
The exponential distribution, with its property of a constant hazard
rate, is frequently used in reliability engineering as a survival model for in­
animate objects such as machine parts (see Chapter 12). Like the uniform
distribution, however, it is not appropriate as a model for human survival
over a broad range, but is used extensively over short intervals, such as one
year, due to its mathematical simplicity. This will be explored in Section
J.5.2.
Since we do not contemplate using the uniform or exponential as a
model for human survival, we use T, rather than X for our failure time
random variable. For the next three distributions, we use X to suggest that
they are more useful as models of human survival.
2.3.3 The Gompertz Distribution
This distribution was suggested as a model for human survival by Gompertz
(33] in 1825. The distribution is usually defined by its hazard rate as
>.(x) = Bc X , x 2 0, B > 0, c > I.
(2.31)
Then the SDF is given by
(2.32)
The PDF is given by >.(x) · S(x), and is clearly not a very convenient
mathematical form. In particular, the mean of the distribution, E [X], is not
easily found.
Examples of Parametric Survival Models
21
2.3.4 The Makeham Distribution
In 1860 Makeham [53] modified the Gompertz distribution by taking the
HRFto be
.\(x)
= A +BcX, x?: 0, B>O, c > 1, A> -B.
(2.33)
Makeham was suggesting that part of the hazard at any age is independent of
the age itself, so a constant was added to the Gompertz hazard rate.
The SDFfor this distribution is given by
(2.34)
Again it is clear that the PDF for this distribution is not mathematically
tractable, so the calculation of probabilities, moments, or other quantities i-s
somewhat difficult.
2.3.5 The Weibull Distribution
This distribution is defined by
.\(x)
= k · x n , x ?: 0, k > 0, n > - l.
(2.35)
Its SDF is given by
x
S(x)
= exp [ - fo k · y n dy] = exp [ - \
�n� I ].
(2.36)
2.3.6 Other Distributions
Other probability distributions are very useful as models for other random
variables, such as the amount of claim random variable in non-life insur­
ance applications (see, for example, Hogg and Klugman (37j). These distri­
butions, which include the gamma, the chi-square (a special case of the gam­
ma), the normal, the lognormal, the Pareto, and others, are not appropriate
for the failure time random variable which we are considering in this text.
The chi-square distribution, however, is useful in testing the fit of
empirical data to a hypothesized parametric distribution (see Chapter·8).
22
The Mathematics of Survival Models
2.3. 7 Summary ofParametric Models
We have briefly explored five distributions here: two (uniform and expo­
nential) which are mathematically simple, and three (Gompertz, Makeham
and Weibull) which are not.
For our actuarial survival model, denoted by S(x), the last three will
receive further consideration in Chapter 8. For many illustrations, where we
wish to avoid mathematical complexity, we will use the uniform or the expo­
nential for iliustrative purposes only, not necessarily suggesting that they are
applicable in practice. The exponential has been assumed to be applicable in
many situations not involving healthy human lives, and has been widely used
in those situations.
2.4 CONDITIONAL MEASURES AND TRUNCATED DISTRIBUTIONS
Thus far we have only considered probabilities measured from age x = 0,
denoting such probabilities by S(x) or F(x). Specifically, such probabilities
were unconditional, since we knew only that the person was alive at x = 0.
Now we consider the case of a person known to be alive at age x > 0, and
we seek probabilities (and densities) of survival (or failure) measured from
age x.
2.4.1 Conditional Probabilities and Densities
What is the probability that a person, known to be alive at age x, will still be
alive n years later (i.e., at age x+n)? We seek
Pr(survival to x+n, given survival to x).
If we multiply this conditional probability by the probability of obtaining the
condition, which is S(x), we obtain the unconditional probability of survival
to age x+n, which is S(x+n). Thus the desired probability, which we denote
by nPx, is given by
11Px =
S(x+n)
S(x) .
(2.37)
The companion conditional probability for death prior to age x+n,
given alive at x, is given by
=
.
( 2.38)
23
Conditional Measures and Truncated Distributions
It is important to distinguish nPx , a conditional probability, from the
unconditional probability represented by S(n;x). In each case we seek the
probability that a person age x will survive to age x+n. When we determine
this probability in accordance with the model S(x), it is conditional, it is denoted by nPx , and it is given by S�(!;). If the desired probability is deter­
mined from S(t;x), then it is unconditional, it is given directly by S(n;x), and
it is denoted by nP[x}, to distinguish it from nPx·
Similar remarks hold for the companion probability of death prior to
age x+n. If it is determined from S(x), it is conditional (on survival to x), it
is given by S(x) �(�j+n), and it is denoted by nqx. But if this probability is
determined from S(t;x), then it is unconditional, it is given directly by F(n;x),
and it is denoted by nq[xJ·
This is not to suggest that we cannot have conditional probabilities
in terms of S(t;x), as shown by the following example.
I EXAMPLE 2.2 I! Find, in terms of S(t;x), the probability that a person se­
lected at age x, but known to be alive at x+ 1 0, will die prior to age x+20.
I SOLUTION I We seek the probability of death prior to age x+20, given
alive at age x+ IO. We denote this probability by 1oq[x)+IO· It is equal to
1-Pr(Survival to x+20, given survival to x+lO) = l -10P[x)+IO· Now if
the conditional probability 10P(x}+10 is multiplied by the probability of
obtaining the condition, which is S(lO;x), the result is the unconditional
probability of survival to x+20, namely S(20;x). Thus
_ 1 - S(20;x)"
- 1 - IOP[x}+IO 1oq[x}+IO _
S(lO;x)
D
Consider next the PDF for death at· age y, given alive at age x, for
y > x. If this conditional density is multiplied by the probability of obtaining
the condition, which is S(x), then the unconditional density, which is f(y),
results. Thus the conditional density is
.fu2
S(x)"
(2.39)
We will derive this conditional density more formally in the next subsection.
Finally, consider the conditional HRF (or force of mortality) for
death at age y, given alive at age x (y > x). Recall that the HRF is itself
always conditional on survival to the age at which it applies. (There is no
The Mathematics of Survival Models
24
such thing as an unconditional HRF.) Thus, since y > x and µy itself is
conditional on survival to y, then the statement "given survival to x" is
redundant. Therefore, this "conditional" HRF to which we allude is clearly
the same as µy itself. This intuitive result will be shown more formally in the
following subsection.
2.4.2 Lower Truncation ofthe Distribution ofX
When we speak of probabilities (or densities) conditional on survival to age
x, we. are dealing with the distribution of a subset of the sample space of the
random variable X, namely those values of X which fall in excess of x. This
distribution is called the distribution ofX truncated below at x.
Our conditional survival probability nPx can now be stated formally
as
(2.40)
nPx = Pr(X>x+n IX>x) = S(x+n IX>x).
In words, this asks for the probability that the age at death will exceed x+n,
given that it does exceed x. It is easy to see that this is the same concept as
"probability of survival to x+n, given survival to x." Thus, from Equations
(2.37) and (2.40) we find that
S(x+n IX> x)
= S(x+n)
(2.41)
S(x)
Similarly,
= Pr(x < x:::; x+nlX>x) = F(x+nlX>x).
(2.42)
Comparison of Equations (2.38) and (2.42) shows that
S(x+n)
F(x+n IX>x) - S(x) S(x)
F(x+n) - F(x)
1 -F(x)
(2.43)
since S(x) = 1 - F(x). Note that both (2.41) and (2.43) result from the
general probability relationship P(AfB) · P(B) = P(A n B).
Next, the conditional density function for death at age y, given alive
at age x (Y > x), is denoted by /(y IX>x). We have
f(yiX>x)
=
d
dy F(yJX>x)
d F(y) - F(x)
I -F(x)
= dy
=
f(y)
l _; F(x)'
Conditional Measures and Truncated Distributions
25
since! F(x) = 0. Thus we have
f(ylX> x)
j(y)
S(x)'
(2.39a)
as already established intuitively by (2.39).
Finally, the alleged "conditional HRF at y, given alive at x (y > x),"
was shown intuitively to be the same as the basic µy (or A(y)). This result
can now be mathematically verified. Denoting this "conditional HRF" by
A(yfX>x), we have
_ f(ylX> x) _ j(y) . S(y) _ flfl _
.X(y lX> x) - S(ylX> x) - S(x) 7 S(x) - S(y) - .X(y). (2.44)
In summary, the functions S(ylX> x), F(ylX>x) andf(ylX> x)
are the functions for the truncated distribution of X, truncated below at x.
The HRF for this truncated distribution, denoted by .A(y IX> x), is identical
to the untruncated A(y).
2.4.3 Upper and Lower Truncation of the Distribution ofX
A more general view of truncated distributions is to consider the distribution
of the subset of the sample space of X which falls between y and z. Still
using X for the age at death random variable, the truncated SDF is given by
S(xjy<Xsz) - Pr(X>xly<XSz)
Pr(x < XS zly < Xsz),
(2.45)
s
for y < x z. In words, we speak of the probability that death will occur
after age x, given that it does occur between y and z. Since it must occur
prior to z, then we are really talking about death between x and z. If this con­
ditional probability is multiplied by the probability of obtaining the condi­
tion, which is S(y) - S(z), then the unconditional probability for death be­
tween x and z, which is S(x)-S(z), results. Thus
(x)-S(z)
S(xly<XSz) = S
S(y)-S(z)"
(2:46)
26
The Mathematics of Survival Models
The corresponding truncated CDF is given by
F(x !Y <XS z)
= Pr(y <XS x IY < X � z).
(2.47)
SinceF(xJy<X�z) = 1-S(xly<X�z),we have
F(x I Y <X
) - S(x) _ F(x)- F(y)
s z) = S(y
S(y) - S(z) - F(z)-F(y)'
(2.48)
directly from Equation (2.46).
Next, the doubly-truncated PDF is given by
d
d S(x)-S(z)
f (xlY < X � z) - - dx S(xly < X s z) = - dx S(y)-S(z)'
producing
f(x IY < X
Jx
s z) = S(y)/(x)
_ S(z)'
(2.49)
Jx
since - S(x) = f(x), and - S(z) = 0.
Finally, the doubly-tnmcated HRF is given by
A(xly<Xsz)
producing
A(xly <Xs z)
(xly<X:Sz)
= fS(x
Xsz)
Jy<
'
_,_ S(x) - S(z) _
f(x)
f(x )
.
S(y) - S(z) - S(x)-S(z)"
- S(y) -S(z)
Sincef(x) = A(X) · S(x) in the untruncated distribution, then we have
A(x IY < X � z)
A(x) · S(x)
= S(x)
-S(z)"
(2.50)
Equation (2.50) shows that, whereas truncation only from below did not
affect the HRF, truncation from above does (since the truncated HRF is a
function of z). This result is intuitive. Since the HRF at x is conditional on
survival to x, truncation below x is immaterial. However, truncation above x
has an effect on the HRF at x, since the time interval remaining for death is
shortened. It should be clear: that as z ---,. x from above, A(x I y < X:::; z)
becomes infinitely large, since the interval for death approaches zero. This
result is seen mathematically by taking the limit as z ---,. x in Equation (2.50).
Conditional Measures and Truncated Distributions
27
Z.4.4 Moments of Truncated Distributions
The first moment of the doubly-truncated distribution of Xis given by
E[Xly < X� z]
z
= l x · f(xly < X� z)dx.
(2.51)
Of special interest is the distribution of X truncated only below at y. Then
E[XIX> yJ
00
= £ x · f(xlX> y)dx,
(2.52)
if the expectation exists. Since Xis the age at death of a person known to be
alive at y, then Equation (2.52) gives the expected age at death for such a
person. If we subtract y from this expected age at death, we obtain the expec­
ted future lifetime of such a person. This expected future lifetime is denoted
by �y, and is called the expectation of life at age y. Formally,
�Y
= E[XIX> y] -y.
(2.53)
00
Since J;, f(x IX> y)dx = 1, we can write
00
�Y = £ (x-y) · f(xlX> y)dx
= fo t · J(t+y(X> y)dt,
(2.54a)
00
(2.54b)
and sincef(t+ylX> y) is the PDF of (X-ylX> y), then "expected future
lifetime" is a good name for �yFurthermore, if it exists, then
E[X 2 IX> y]
=
loc
x 2 • f(x IX> y)dx.
(2.55)
Then the variance of future lifetime is given by
2
Var(X-y!X> y) = Var(XIX> y) = EIX2 IX> yJ - { E[XIX> y]
(2.56)
}.
The Mathematics of Survival Models
28
2.4.5 The Central Rate
Another type of conditional measure over the interval from age x to age x+ I
is called the central rate of death, and is denoted by mx . It is defined as the
weighted average value of the HRF ,\(x) over the interval, using, as the
weight for .\(y), the probability of survival to age y. Formally,
fx + i S(y). >.(y) dy
x
rx+I
.
Y
Jx S(y)d
(2.57)
,
where the denominator is the sum of the weights for a continuous case
weighted average.
More generally, nmx is the average hazard, or central rate of death,
over the interval from x to x+n, ancl is given by
m
n x -
S(y). >.(y)dy _c+n S(y)dy
x+n
fx
J; S(x+s) · ,\(x+s)ds ,
fon S(x+s)ds
(2.58)
the second expression resulting from the simple change of variable y = x+s.
If we divide both numerator and denominator of (2.58) by S(x), we obtain
rn S(x+s) .
Jo
>.(x+s).ds
=
rn S(x+s) d
Jo S(x) s
S(x )
since sPx is the conditional probability
J; sPxµx+s ds ,
J; sP ds
x
(2.59)
s�;t, and µ +s is the standard actu­
x
arial symbol for >.(x+s). The second expression in (2.59) is a common ac­
tuarial form for nmx. We will return to this function in the next chapter, and
make some use of it in estimating the tabular survival model in Part II of the
text.
!!EXAMPLE 2.31! If X has an exponential distribution, show that this im­
plies mx = - lnpx ,
I SOLUTION I IfXis exponential, the HRF is constant, with >.(y) = >. for
>.. f+t S(v) dy =
.
all y. Then, from Equation (2.57), mx = x+i
>.. Furthermore,
J,
S(v)dy
x
e--'(x+t)
S(x+l)
we also have Px = S ) = r-'<x> = e-A . Thus). = -lnpx , and since
(x
m.-.: = >., then mx = -lnpx.
D
29
Conditional Measures and Truncated Distributions
2.4. 6 Use of Conditional Probabilities in Estimation
We have noted that the main business of this text will be the estimation of an
operative survival model, such estimation to be based on the data of a
sample.
Suppose we wish to estimate, say, S( I 0), the probability of survival
from t = 0 tot= 10. In many cases the nature of the study (and the data)
will suggest that we consider only the time interval from t = i to t = i+ 1,
and estimate the conditional probability of survival over that interval. That
is, we will estimate s�t)l), the probability of survival to i+ 1, given alive at
i. This conditional probability has been called p;, so the estimate of it which
we obtain will be called
for i = 0, 1, ... , 9. We will then obtain our
estimate of S(l 0) by multiplying these several
Thus we will obtain
A
A
A
A
•
S( l 0) = Po · p 1 • • • • • p9, or, m general
P;,
A
S(t)
P;·
= Po · Pt · · · · · P,-1·
A
A
(2.60)
A
In many cases it will be natural to first estimate q;, the conditional proba­
bility of failure (death) in (i, i+ 1], given alive at i; then take
= 1A
where is the estimate of q;; and finally obtain S(t) by multiplying these
conditional estimates. This approach to estimating the survival function
will be utilized mainly in Chapters 6 and 7.
q;
P;
q;,
P;
2,5 TRANSFORMATIONS OF RANDOM VARIABLES
Suppose we have a random variable X, with known probability distribution,
and we consider a new random variable Y, which is some function of X.
That is, Jet
y
= g(x)
(2.61)
be a function of x such that the inverse function x = g- 1 (y) = h(y) exists.
We seek the probability distribution of .Y.
Let y = g(x) be a strictly increasing function, as shown in Figure 2.1
on page 30. Since g(x) is increasing, then if Xis Jess than or equal to x, it
follows that Y is less than or equal to the unique value of y which corres­
ponds to the given value of x. Thus if X � x, then Y � g(x). Conversely, if
Y � y, then X � h(y), and the probabilities of these events are equal.That is,
The Mathematics of Survival Models
30
Pr(Y � y)
= Pr(X :S h(y)),
(2.62)
or
F(y) = F[h(y)].
(2.63)
Equation (2.63) can be confusing since the CDF's on opposite sides of the
equation are not the same function. The one on the left is the CDF of the
random variable Y, whereas the one on the right is for the random variable X
To clarify this we write
Fy(y)
= Fx[h(y)].
(2.63a)
y = g(x)
y
x
!FIGURE 2.11
from (2.63a), which relates the CDF's of the random variables X and
Y, we can derive relationships for the SDF's, PDF's and HRF's as well.
Since the SDF is the complement of the CDF, it follows from (2.63a)
that 1 - Sy (y) = 1 - Sx[h(y)], or that
Sr(y)
= Sx[h(y)].
(2.64)
Next, the PDF is the derivative of the CDF, so we differentiate both
sides of (2.63a) with respect toy, obtaining
fy(y)
d
d
d ( )
= dy
Fr y = cly Fx[h(y)J = fx[h(y)] · dy h(y),
using the chain rule to differentiate Fx[h(y)], Since h(y) is simply x, we can
write
dx
(2.65)
fy(y) = fr[h(y)] . dy"
Transformations of Random Variables
31
Finally, the HRF is the ratio of the PDF to the SDF. Thus
' (y) -
"Y
-
jy(y)
Sy(y)
dx
= f Sx[h(y)·]i = Ax[h(y)] · dy"
x[h(y)]
(2.66)
[EXAMPLE 2.4[1 Suppose X has an exponential distribution with >. = 1.
Lety = g(x) = x 112 . Find the SDF, PDF and HRF of the transformed random
variable Y.
[SOLUTION[ Note that y = g(x) is strictly increasing, and x = h(y) = y 2 .
Then Fy (y) = Fx(y 2 ). Since Xis exponential, then Fx(x) = 1-e-x, so we
2
2
2
have Fy (y) = I - e-Y , and Sy (y) = e-Y • Next,fr(y) = Fy (y) = 2y · e-Y •
2
2
Finally, >. y (y) = 1 .{rfrri) = :·�;; = 2y. Alternatively, since we have
= 2y. Note that
>.x(x) = >.x[h(y)] = l, then, from (2.66), ,\y(y) is simply
Y has a Weibull distribution with k = 2 and n = 1.
0
t
!
If y = g(x) is a strictly decreasing function, as shown in Figure 2.2,
then our reasoning and our results change a bit.
y
y = g(x)
x
!FIGURE 2.2!
Here we can see that if Xis less than x, then Y will be greater than the value
of y which corresponds to the given value of x; or, conversely, if Y > y, then
X < h(y). In terms of probabilities, we have
Pr(Y > y)
= Pr(X < h(y)) = Pr(X:::; h(y))
(2.67)
32
The Mathematics of Survival Models
or
Sy(y) - Fx[h(y)J - 1 - Sx[h(y)J.
-
(2.68)
Then
I - S y (y) - Sx [h(y)],
- 1 - Fx [h(y)] -
(2.69)
:/y F (y) = - :/y F [h(y)] = -fx[h(y)] · �;,
(2.70)
Fy(y)
and
jy(y)
=
x
r
by the chain rule. Since x = h(y) is a decreasing function, then : is negative
andfr(v) is positive, as required. Finally,
' '') - fy(y)
-"Yv'
S y (V) -
-fy[h(y)]. *
Sx [h(y)]
dx
1 - Sx [h(y)J - - .\x[h(y)] · 1 - Sx [h(y)J · dy·
(2.71)
Note that the expression for .\ y (y) in terms of .\ x (x) is not as convenient as it
is for the case where y = g(x) is a strictly increasing function.
l!EXAMPLE__jJJI As in Example 2.4, let X have a standard exponential
distribution, so that Fx(x) = l - rx , bl!t this time let y = g(x) = �. Again
find the SDF, PDF and HRF of Y.
have x = h(y) = �, so Sr(Jl) = Fx (h(y)] = l-e-< 1 !1)_
Next we have f (y) = - �' S (y) = y-2 · e-(l/y)_ Finally, the HRF is
[SOLUTION) We
y
y
(y)
.\ y (y) - fr
Sy (y)
fy)
= y-2 . l -e-<t-<t
f .
e
y)
Again note that with .\x (x) = 1 and : = -- y-2, the last expression of
Equation{2.71) can.be used to find .\ y (y).
0
Mean and Variance of Transformed Random Variables
33
2.6 MEAN AND VARIANCE OF TRANSFORMED
RANDOM VARIABLES
If Y = g(X) is a random variable transformed from the random variable X,
we may wish to find the expected value and variance of Y. Although/r(y) is
obtained in the transformation, it might tum out that the antiderivative of
y ·f(y) does not exist, so E[Y) cannot be found in that manner. Note that
this is the case in both Example 2.4 and Example 2.5, although E[Y] in Ex­
ample 2.4 can still be found, as we will see in the exercises.
Furthermore, the ability to find the mean of Y directly from the mean
of X, and the variance of Y directly from the variance of X, is limited to
linear transformations of X to Y. That is, if Y = aX+b, then we have
E[Y] =a· E[X]+b, and Var(Y) = a 2 • Var(X). But if Y = x 1 !2 or
Y = x- 1, as in our two examples, no such convenient result can be found.
For such cases we could use an approximate approach to finding E[Y]
and Var( Y). One such approach would be the use of approximate integration
to evaluate
y2 -f(y)dy. Another approach is the method
y-j(y)dy or
of statistical differentials, which requires that E[X] and Var(X) be known,
and that y = g(x) be twice differentiable.
Let E[X] = m. We begin by expressing g(X) in a Taylor series, ex­
panded about X = m. Thus
It
It
g(X) = g(m) + (X-m) · g'(m) + f(X-m)2 · g''(m) + · · ·,
(2.72)
where g'(m) and g''(m) are short-hand notation for g'(x) lx==m and for
g''(x) lx=m , respectively.
Next we take the expected value of both sides of (2. 72), truncating the
Taylor series after its third term, obtaining
E[Y]
= E[g(X)] � g(m) + !g''(m) · Var(X),
(2.73)
sinceE [g(m)] = g(m),andE[X-m] = 0.
A rougher approximation is given by truncating the Taylor series after
only two terms, obtaining
E[g(:x)] � g(m).
(2.74)
The variance of g(X) is the expected value of {g(X) -E[g(X)]} 2.
Using only the first two terms of (2.72) for g(X), and using (2.74) for
E[g(X)], we have g(X)- E[g(X)] � (X-m) · g'(m). Then
34
The Mathematics of Survival Models
= E[ { g(X) - E[g(X )]
Var[g(X )] --
r]
:=:::!
E[(X-m)2 · [g'(m)] 2 )
= [g'(m)]2 · Var(X ).
(2.75)
!!EXAMPLE 2.611 Show that Equations (2.74) and (2.75) are exact for the
linear transformation Y = g(X) = aX+b.
!SOLUTION[ We know that E[Y] =a· E[X]+b = am+b = g(m), so
(2.74) is exact. Furthermore, Var(Y) = a2 · Var(X). Since g'(m) = a, then
[g'(m)] 2 = a2, so (2.75) is exact. [Note: It is easy to see that the approxi­
mations will be exact for linear transformations, since g''(x) and all higher
order derivatives are zero.]
0
A major use of these results will arise in Part II of the text, where we
wish to find the variance of an estimator. Frequently that estimator will be a
complex function of another random variable, whose mean and variance are
known (at least approximately), so the method of statistical differentials can
be used.
In some cases, we may encounter a random variable which is a func­
tion of two (or more) other random variables, and we seek its mean and vari­
ance. For example, if Y = g(Xi, X2), and if E[Xi] = mi, E[X2] = m2,
Var(X1 ), Var(X2 ), and Cov(X 1 ,X2 ) are all known, then we can proceed
similarly to the univariate case.
Expand g(X1, X2) in a bivariate Taylor series as
g(Xi , X2) = g(m1 ,m2) + (Xi-m1) · ii (mi,m2) + (Xr m2) · gz(m1,m2)
+ !(X1-m1)2 · g'{(m1,m2 ) + i(X2-m2 )2 · g!J(m1,m2 )
(2.76)
+ (X 1 -mi) (X2-m2) · g'{,i (m1,m2) + · · ·,
where &(m1,m2) represents
g'/(m, ,m2) represents
"'82
!. g(x1,x2)lx,=mp for i = 1,2. Similarly,
!� g(x1 ,X2)lir::�' i = 1,2, and ii',i<mi , 2) represents
I
=
X2 m2
m
. g(xi ,x2)lx, =mi. Taking the expected value of both sides of (2. 76), we
UX1X2
obtain
x2=m2
E[g(X1,X2) ] � g(m i,m2) + }g'{(m1,m2 ) · Var(X1)
+ fg'{(mi,m2) · Var(X2 ) + ii',2 (m1,m2) · Cov(X1, X2).
{2.77)
35
Mean and Variance of Transformed Random Variables
Note that the expected value of each of the two first-order terms is zero.
A rougher approximation is simply
(2.78)
For the variance of g(X 1 , X2 ), we choose to retain only the first three
terms of (2.76) for g(X1, X2), and use (2.78) for E[g(X 1 , X2 )]. Then
Var[g(X1, X2)] = E[{g(X1, X 2 ) - E[g(X1, X2)]r]
2
� E[{ (X1 -m 1 ) · ii (mi, m2) + (X 2 -m 2 ) · ii(m1,m2)} ]
- E [(X1-m,) 2 [ii(m1,m2)] 2 + (X2 -m2) 2 (ii(m1,m2)] 2
+ 2(X1-m1)(X2-m 2 ) · iJ(m1,m 2 ) · ii(m 1 ,m2)]
-
(ii(m1,m2)] 2 · Var(X1) + (ii(m1,m2)] 2 · Var(X2)
+ 2 · iJ(m1,m 2 ) · ii(m 1 ,m 2 ) · Cov(X1, X 2 ).
(2.79)
If X1 and X 2 are independent or uncorrelated, then the third term in (2.79) is
zero. Extension to three or more random variables is analogous to the bivari­
ate case.
2.7 SUMMARY
This chapter has attempted a thorough exploration of the theory and mathe­
matics of survival models, using, for the most part, simply the concepts of
the CDF, SDF, PDF, and HRF of a survival distribution. On occasion, some
properties of the conceptual survival model were illustrated by using specific
examples of continuous survival distributions, such as the exponential,
Gompertz, and so on.
It is very important that the theoretical material of this chapter be
thoroughly understood before dealing with the statistical estimation of sur­
vival models. In particular, the difference between conditional and un­
eonditional distributions and mortality measures is very important, since
conditional mortality probabilities are frequently used to develop the final,
unconditional, survival distribution function.
The Mathematics of Survival Models
36
2.8 EXERCISES
2.1 Introduction;
2.2 The Distribution of T
2-1.
A survival distribution is defined by S(t) = .10(100- t) 1 12, over the
domain O � t � 100. Find each of the following:
(c) A(75)
(d) E[T]
(b) A(50)
(a)f( 36)
(e) Var(T)
2-2.
Suppose the hazard rate of a survival distribution is a linear function,
A(t) = a+bt, where a> 0 and b > 0. Find each of the following: (a)
(c) The mode of the distribution
(b) f(t)
S(t)
2-3.
Suppose a survival distribution is defined by S(x) = ax2 +b, for
0 � x � k. If the expected value of Xis 60, find the median of X
2-4.
Explain why the hazard rate A(t) = e-rr, r > 0, is not appropriate for
a survival distribution.
2.3 Examples of Parametric Survival Models
2-5.
If Tis uniformly distributed over [1,3), find Var(T).
2-6.
Let Xi and X2 be independent random variables. Define the variables
Y =-min(X1,X2 ) and Z = max(X1,X2).
(a) Show that the SDF of Yis the product of the SDF's of X1 and X2•
(b) Show that the CDF of Z is the product of the CDF's of X1 and X2 .
(c) Show that if X1 and X2 both have exponential distributions, then Y
has an exponential distribution, but Z does not.
2-7.
Suppose the independent random variables X1 and X2 both have
exponential distributions, with parameters A 1 and .\2 , respectively,
where AJ > A2 . Let Y and Z be defined as in Exercise 2-6. If it is
known that at time t = 2, the SDF of Y has the value .24 and the SDF
of Z has the value .86, then find the value of A i .
2-8.
mlqo = S(m) - S(m+J) is the probability that an entity existing at
time O will fail between t = m and l = m+ l. Determine whether mlqo
is an increasing, decreasing, or constant function of m for each of the
following distributions:
(a) Uniform
(b) Exponential
(c) f(t) = .00125!, 0 � t � 40
Exercises
2-9.
The distribution f(x)
= 2,12fcr12) x<r/2)-I e-x/2, x > 0, is the Chi­
square distribution with r degrees of freedom (r > 0).
(a) Find each of the following for this distribution:
(i) E[X] (ii) Var(X) (iii) E[X- 1]
(b) Find the hazard rate, ).(x), for this distribution if r = 2.
2.4 Conditional Measures and Truncated Distributions
2-10. Show that an exponential distribution truncated from below is still an
exponential distribution.
2-11. Let S(x) = ax+b, 0 :S x :S k. If it is known that 2 p0 = . 75, find 2 m3•
2-12. The survival distribution for a certain laboratory unit known to exist
at t = 5 is given by an exponential distribution truncated above at
t = 15, with parameter ). = .02. Find the median future lifetime of
this unit.
2-13. If S(x) = I - (.0Ix)2 , 0 :S x :S 100, find the expected future lifetime
at the median age.
2-14. Find the hazard rate ).(x IY::; X :S z) for a doubly-truncated exponen­
tial distribution. Why is the HRF not a constant?
2-15. The Type I Least Value distribution is defined by its SDF as
S(x)
x
= exp [ 1:: ], -oo < x < oo.
Show that this distribution, when truncated below at x = 0, be­
comes a Gompertz distribution.
(b) How does the HRF for the untruncated distribution compare
with the HRF for the Gompertz distribution?
(a)
2.5 Transformations of Random Variables
2-16. Let X have a Weibull distribution, and let Y = lnX Find the SDF
and HRF of Y. Can you give a name to this distribution?
2-17. Show that the transformationy = Fx(x), where Fx(x) is the (strictly
increasing) CDF of X, leads to the result fy(y) = 1, so that Y has a
standard unifonn distribution.
The Mathematics of Survival Models
38
2-18. If fx(x) = 1r(l�x2), -oo < x < oo, find the distribution of Y = X 2 .
2.6 Mean and Variance of Transformed Random Variables
2-19. Consider the transformed random variable Y defined in Example 2.4.
(a) Find E[Y] and Var(Y) directly fromjy(y). [Hint: For E[Y], sub­
u 1 12 • e-u du = I'(l .5).]
stitute u = y 2 , and recognize that
(b) Approximate the mean and variance of Y by the method of statisti­
cal differentials; compare them with the exact values from part (a).
ft
2-20. Approximate the mean and variance of the transformed random vari­
able Y defined in Example 2.5. [Note: This exercise shows how bad
the approximations from the method of statistical differentials can be,
since, in fact, E[Y] = E[X- 1 ] does not exist.]
2-21. Let X be normally distributed with mean µ and variance a2 .
(a) Approximate the variance of Y = e x.
2
(b) Since Y has a lognormal distribution, then E[Y] = eµ+o- /2 and
E[Y 2] = e2µ + 20- • If Var ( Y) is expressed as a series in a2 , and
then truncated to agree with the approximate result in part (a),
what is the first neglected term.
2
2-22. Two independent random variables X1 and X2 have PDF's given by
Ji( x1) = l2x1(1-x1), 0 S x1 S I, andh(x2) = 2 x2, 0 S x2 S I. The
product Y = X1 X2 has PDF jy(y) = 12y(l-y)2, 0 Sys I. Calculate
the exact mean and variance of Y, and compare them with the approxi­
mate values obtained by the method of statistical differentials.
CHAPTER3
THE LIFE TABLE
3.1 INTRODUCTION
In this chapter we describe the nature of the traditional life table, showing
that it can have all the properties of the survival models described in Chapter
2. Again we stress that a total understanding of the structure and properties
of this model is essential for our main topic of model estimation.
When a survival model is presented in the life table format, it is cus­
tomary to use notation and terminology which differ somewhat from that
presented in Chapter 2. A major objective of this chapter will be to show
clearly the correspondence between notation used in the conceptual model
and that used in the specific life table model.
The reader should realize that life tables were developed by actu­
aries independently from (and a century earlier than) the development of the
statistical theory of survival models as probability distributions, and the esti­
mation thereof. For this reason, traditional life table notation and termin­
ology will not tend to reveal the stochastic nature of the model as clearly as
is done by the conceptual model in Chapter 2. By showing the corres­
pondence of the life table symbols to those of the conceptual model, we
intend to correct this.
The life table was defined in Section 1.3 as a table of numerical
values of S(x) for certain values of x. Table 3.1 illustrates such a table.
TABLE3.1
x
S(x)
0
1.00000
1
.97408
.97259
2
3
.97160
4
.97082
..
..
.
.
.00001
109
110
.00000
The Life Table
40
Typically a complete life table shows values of S(x) for all integral
values of x, x = 0, l ,. . . . Since S(x) is represented by these values, it is clear
that a practical upper limit on x must be adopted beyond which values of
S(x) are taken to be zero. Traditionally we use w for the smallest value of x
for which S(x) = 0. That is, S(w-1) > 0, but S(w) = 0. In Table 3.1,
w = I!O.
From Table 3. I, one could calculate conditional probabilties repre­
sented by nPx and nqx for integral x and n. However, these are the only
functions which can be determined from the tabular model. Functions such
as/(x), >.(x), mx , and �x cannot be determined from the tabular model unless
we expand the model by adopting assumed values for S(x) between adjacent
integers. We will pursue this in later sections of this chapter.
j!EXAMPLE 3.t!I From Table 3.1, calculate (a) the probability that a life
age O will die before age 3; (b) the probability that a life age 1 will survive to
age 4.
:m
!SOLUTION!
(a) This is given directly by F(3) = 1 - S(3) = .02840.
= .99665.
(b) This conditional probability is given by 3PI =
D
3.2 THE TRADITIONAL FORM OF THE LIFE TABLE
The tabular survival model was developed by the early actuaries many years
ago. The history of this model is reported throughout actuarial literature,
and a brief summary of this history is presented by Dobson (22].
Traditionally, the tabular survival model differs from Table 3.1 in
two respects. Rather than presenting decimal values of S(x), it is usual to
multiply these values by, say, I00,000, and thereby present the S(x) values as
integers. Secondly, since these integers are not probabilities (which S(x)
values are), the column heading is changed from S(x) to ex , where£ stands
for number living, or number of lives. Thus did the tabular survival model
become known as the life table.
Since S(O) = l , then clearly £0 is the same as the constant multiple
which transforms all S(x) into ex . This constant is called the radix of the
table. Formally,
(3.1)
fx = P.o • S(x).
Using a radix of 100,000, we transform Table 3.1 into Table 3. la.
41
The Traditional Form of the life Table
TABLE3.la
x
lx
0 100,000
I
97,408
2
97,259
3
97,160
4
97,082
..
.
109
110
..
.
I
0
The basic advantage of the traditional form of the life table is its suscepti­
bility to interpretation. If we view 4, = 100,000 as a hypothetical cohort
group of new-born lives, then each value of fx represents the survivors of
that group to age x, according to the model. This is a convenient, determin­
istic, interpretation of the model. Of course, since fx = £0 • S(x), and S(x) is
a probability, then fx is the expected number of survivors to age x out of an
original group of £0 new-horns. This connection between S(x) and fx is also
given in Chapter 1 ofJordan [41], and in Chapter 3 of Bowers, et al. (12].
The distinction between S(x) and fx is sufficiently subtle to justify a
second look at them. We first recognize that mathematically the two ftmc­
tions are identical. They embody the same information. The basic difference
between them lies in their interpretation. S(x) is a probability function, and
therefore, has a probabilistic interpretation. fx is interpreted as a number of
persons living at age x (out of an original cohort of fo), and thus has both a
probabilistic and a deterministic interpretation.
Although the basic representation of the tabular survival model is in
terms of the values of fx , it is customary for the table to also show the value
of several other functions derived from fx . We define
(3.2)
or, more generally,
(3.3)
Since ix represents the size of the cohort at age x, and fx+n is the number of
them still surviving at age x+n, then clearly nd:x. gives the number who die
between ages x and x+n. (This portrayal of number dying might be the
reason for the frequent historical reference to these models as mortality
tables.) Furthermore,
42
The Life Table
(3.4)
or, more generally,
(3.5)
gives the conditional probability of death, given alive at age x. Finally, we
have
(3.6)
as the conditional probability of surviving to age x+n, given alive at age x.
With n = l, we have the special case
_ fx+I
Px - fx
(3.7)
Recall that the conditional probabilities nPx and nqx were defined in
Section 2.4 in terms of S(x). The consistency of those definitions with the
ones presented in this section is easily seen since fx is simply £0 • S (x). We
redefined nPx and nqx in terms of fx here simply to complete our description
of the life table.
IIEXAMPLE 3.2ll From Table 3.la, find (a) the number who die between
ages 2 and 4; (b) the probability that a life age l will survive to age 4.
[SOLUTION!
(a) This is given by 2d2 = f2 - £4 = 177.
= .99665. (Compare with part (b) of Example
(b) This is given by 3p1 =
3.1.)
t
0
3.3 OTHER FUNCTIONS DERIVED FROM ix
Although a life table only presents values of fx for certain (say, integral)
values of x, we wish to adopt the view that the fx function which produces
these values is a continuous and differentiable function. In other words, we
assume that a continuous and differentiable fx function exists, but only
certain values of it are presented in the survival model. The reason we make
this assumption is that there are several other mortality functions which can
be derived from fx if fx is continuous and differentiable.
Other Functions Derived from fr
43
If values of fx are known only at integral x, the question of how to
evaluate these additional functions then arises, and the usual way to
accomplish this evaluation is to make an assumption about the fonn of fx
between adjacent integral values of x. (Recall that we earlier referred to the
assumptions of a form for S(x) as mortality distribution assumptions.)
In this section we will derive these several new functions from fx
symbolically, assuming fx to be continuous and differentiable. In Section
3.5 we will discuss the common mortality distribution assumptions, and
show how they allow us to evaluate the functions of this section from a table
of fx values at integral x only. We will also interpret these mortality distri­
bution assumptions in terms of both fx and S(x).
3.3.1 The Force of Mortality
The derivative of fx can be interpreted as the absolute instantaneous annual
rate of change of fx. Since fx represents the number of persons alive at age
x, then the derivative, which is the annual rate at which lx is changing, gives
the annual rate at which people are dying at age x. This derivative is negative
since lx is a decreasing function. To obtain the absolute magnitude of this
instantaneous rate of death, we will use the negative of the derivative. Final­
ly, since the magnitude of the derivative depends on the size oflx itself, we
obtain the relative instantaneous rate of death by dividing the negative
derivative oflx by fx itself. Thus we have
(3.8)
which we call the force of mortality at age x. Now since lx =lo· S(x), then
we see that (3.8) is the same as
,\(x) =
-1; S(x)
S(x)
f(x)
S(x)·
(3.9)
Thus the hazard rate and the force of mortality are identical.
If we multiply both sides of Equation (2.11) by £0, replace t with x,
and substitute µy for ..\(y), we obtain
(3.10)
44
The Life Table
In the life table context, S(x) = xPo = exp[ - fox µy dy] can be interpreted as
a decremental factor which reduces the initial cohort of size £0 to size fx at
age x.
By a simple variable change we can write Equation (3.8) as
(3.8a)
a form in which the force of mortality will frequently be expressed.
jlEXAMPLE 3.31! Show that the force of mortality is the limiting value of
the probability of death over an interval, divided by the interval length (in
years), as the interval length approaches zero.
!SOLUTION! Consider first a one-year interval, with qx = �:. Then consid. , 12q"
.
.
·. . Now, m genera1 , cons1.der
er a h a If-year mterva 1 with 1/2
= e" 1-12e<+1;2
"
e
£lx9x = e)( - eHilx, and show that Jim ,:),xqJC = µx. We have
Llx
Llx . e
Llx-o Llx
)(
lim[ex-fx+Llx] = I
&--o
Llx · ix
lim[ex-fx+t1x] =
fr . .1.r�o
Llx
.!!... ex] =
1.(ix
dx
µx,
by Equation (3.8).
0
3.3.2 Tlte Probability Density Function ofX
With the force of mortality, which is the same as the hazard rate, now de­
fined, the next function to develop from fx is the PDF of the age-at-death
random variable X (Remember that we wish to show that the life table is a
representation of the distribution of this random variable.)
From Equation (2.7) we have f(x)= ,\(x) · S(x). In the life table
context, we have >.(x) = µx and S(x) = fxlf0. Thus we have
(3.11)
Since, from (3.8), :fx fx = - fxµx , then dividing both sides by £0 gives
(3.12)
45
Other Functions Derived from ix
[EXAMPLE 3.4!j Show that f0 j(x)dx = I.
00
[sOLUTIONI Since J(x) = xPoAx, we have fo xPoµx dx = -xPol�, from
(3.12). Thus we have o Po - ooPo = I, since oPo = I and 00po = 0.
D
00
With the PDF in hand, we can now find E[X], which we recall is de­
. noted by �o- (Throughout this and the following section, all expectations are
assumed to exist.) We have
�o
= E[X] =
1
00 x
· J(x) dx
00
1
=
x · xPoµx dx.
(3.13)
Integration by parts produces the alternative formula
�o
= E[X] =
100
O
xPodx
= fO · 1
O
lfwe define
then we have
�o
= E[X] =
lo·
00
fx dx.
(3.14)
(3.15)
(3.16)
The function T0 is a special case of Tx, defined by
(3.17)
This function Tx is very important. We will encounter it again, and in Sec­
tion 3.3.5 we will give an interpretation of it.
The second moment ofXis found from
(3.18)
Integration by parts produces 2 ·
Equation (3.17),
ft x · xPo dx = fo · ft x · £.� dx. From
(3.19)
The Life Table
46
t·
by the fundamental theorem of calculus, so J fx dx = - Tx. Then integrating
x
fox Tx dx (since the term j� x · ex dx, we obtain
Tx
by parts
evaluates to zero at both limits). We define
ta ·
ta ·
(3.20)
Yo
obtaining
--ro-·
2 · Yo
(3.21)
Finally, from (3.21) and (3 .16), we have
Var(X)
2 · Yo
= E[X 2 ] - {E[X]r = -,;;--
(3.22)
3.3.3 Conditional Probabilities and Densities
We have already discussed the conditional probabilities nPx and nqx in terms
of both S(x) and fx.
Another conditional probability of some interest is denoted by nJmqx.
It represents the probability that a person known to be alive at age x will die
between ages x+n and x+n+m. In terms of the formal probability notation
of Chapter 2, nlmqx = Pr[(x+n) < X < (x+n+m)JX> x]. This can also be
expressed as the probability that a person age x will survive n years, but then
die within the next m years. This way of stating the probability suggests that
we can write
(3.23)
Here mqx+n is the conditional probability of dying between ages x+n and
x+n+m, given alive at age x+n. In tum, nPx is the conditional probability of
surviving to age x+n, given alive at age x. Their product gives the proba­
bility of dying between ages x+n and x+n+m, given alive at age x. In terms
of fx , we have, from (3.6) and (3.5),
(3.24)
Other Functions Derived from lx
f
l@;XAMPLE 3.Sf Show that nlmqx
of this result.
47
= nP - n P.t, and give an interpretation
x
+m
[SOLUTION! Since, from (3.3), mdx+n = fx+n - fx+n+m , then (3.24) bef:i+n - fx+n+m
I 1ty Of
comes nlmqx =
= nPx - n +mPx· s·mce nPx lS. the probab·1·
ex
surviving to age x+n, we can think of it as containing the probability of
surviving to any age beyond x+n. If we remove from nPx the probability of
surviving to x+n+m, which is n+mPx , we have the probability of surviving to
D
x+n, but not to x+n+m, which is nlmqx.
Next we wish to explore the conditional PDF for death at age y,
given alive at age x(y > x). From (2.39), we know this conditional PDF is
Now from (3.11) we have /(y) = · f.yµy , and from
f(yiX> x) =
{�r
(3.1) we have S(x) =
;; . Thus
f(ylX> x)
°t
(3.25)
Lettings = y-x, soy = x+s, we have
(3.26)
where the random variable S denotes the length of future lifetime of a person
alive at age x. The conditional PDF given by (3.26) is a very useful function
for developing other results.
If both numerator and denominator of (3.8a) are divided by fx , we
obtain
d
- ds
sPx
(3.27)
sPx
which shows that
(3.28)
The expectedfuture lifetime of a person alive at age xis given by
(3.29)
48
The Life Table
- sPx (from (3.28)), and
- s · sPx evaluates to zero at both limits. Furthermore, sPx = el,:', so (3.29)
since, in the integration by parts, J sPx µ.-r+s ds =
becomes
1 (,c
fx+s ds
.
fx lo
.
f
1x
1
"°
x
(3.30)
fy dy
from (3.17).
By steps parallel to those leading from(3.18) to (3.21 ), we find
(3.31)
where
Finally, from(3.31) and (3.30), we have
Var(S)
= E[S 2 ] - { E[S]
(3 32)
.
r
(3.33)
!!EXAMPLE iiij A survival model is defined by fx = ��:��, x � 0. De­
termine Var(XIX > x), whereXis the age-at-death random variable.
�OLUTIONI We recognize that the age-at-death random variableXand the
future-lifetime random variable S are related by X = x+S, where x is a
constant. Thus, Var(XIX> x) = Var(SIX> x). From(3.33) this variance
2
2
Thus we first find
is given by ;{x
T_Y
and
- ( r:)
= 1 fy dy = 10,0001 (y+l)-3 dy
x
x
Then Var(XI X > x) = (x+t)-3
(x+I)-'
(
(x+n-2
2(x+l)-3
)2
5000(x+I)-2
3(x+l)2
-4-
D
49
Other Functions Derived from fx
3.3.4 The Central Rate
In Section 2.4.5, the central rate of death was defined by Equation (2.59) as
n
Jo sPx µx+s ds
(2.59)
sPx ds
J:
Since sP:c = �, multiplication of both numerator and denominator of
Equation (2.59) by fx produces
(3.34)
(3.35)
where
(3.36)
If n = 1, we have the one-year interval function
Lx = fo 1 f:c+s ds,
(3.37)
and the central rate over the interval (x, x+ I] is given by
mx =
dx
r·
x
(3.38)
Note that there is a simple relationship between L:c and Tx , as defined
by (3.17). There we saw that
Tx =
so
oo
l
x
f.y dy = i I f.y dy + 1
x+
x+2
x+I
x
= Lx + Lx+ I + .. ·,
Tx
oo
= I:Ly.
)l='X
f.y dy + .. ·
(3.39)
The Life Table
50
We have developed Lx and Tx simply as mathematical concepts. It is
now necessary to give descriptive meanings to them, since these interpre­
tations will be helpful in understanding the use of mx , Tx and Lx in the
estimation of survival models from general population data.
3.3.5 The Concept of Exposure
A survival model, expressed by the SDF S(x), is a probability distribution
with all the properties that such distributions possess. The simple transfor­
mation of Rx = £0 · S(x), where Ro is a constant, expresses the same probabil�
ity distribution. The function Rx is the same function as is S(x), except for the
trivial difference that whereas 1 2'.: S(x) � 0, we now have f0 � fx � 0. Any
function or result that can be derived from S(x) can also be derived from fx.
However, an advantage in using fx, instead of S(x), is the ability to
interpret the values of fx as the survivors of an initial, closed, cohort of
newborn lives of size of R0• Successive values of S(x) are probabilities,
which are somewhat abstract, especially to non-mathematicians. But values
of fx have a "real world" meaning, notwithstanding the fact that we are deal­
ing with hypothetical situations.
In turn, the interpretive nature of fx allows for concrete (albeit hypo­
thetical) interpretations of several functions derived from fx . Of particular
usefulness is the function Lx , defined by (3.37).
Recall, from (3 .26), that sPx µx+s is the PDF for death at age x+s,
given alive at age x. If we multiply this PDF bylx , which we interpret as the
number of persons alive in a group at age x, we obtain fx+sµx+s, which is the
rate of deaths occurring in the group at exact age x+s. In tum, fx+s µx+s ds is
the differential number of deaths occurring at exact age x+s. Then
s · fx+sµx+s ds is the total number of years lived by those deaths after attain­
1
ing age x. Finally, 0 s · fx+sµx+s ds gives the aggregate number of years
lived, after age x, by all those who die between age x and age x+ I.
Most of the Rx group, of course, survive to age x+ I, fx+ 1 being the
number who do so. Each of these persons live one year from age x to age
x+ I, so f.x+ 1 also represents the aggregate number of years lived, between
ages x and x+ 1, by those who survive to age x+ 1. Together,
f
(3.40)
gives the aggregate number of years lived between ages x and x+ 1 by the ix
persons who comprised the group at age x,
51
Other Functions Derived from fx
This quantity is measured in units of life-years, and is called expo­
sure, since it is a measure of the extent to which the group is exposed to the
risk of dying. Note that persons who do die contribute to our exposure
aggregation only up to the time of death. This is an important point, since in
Chapter 6 we will encounter two different meanings of the exposure concept,
where this rule will not hold.
We can simplify (3.40) by performing integration by parts, to obtain
fx+I -S · fx+si: +
l
l
l fx+sdS = l f + ds,
x s
which has already been defined in (3.37) to be lx.
This result suggests the possibility of directly. interpreting Lx , as
defined by (3.37), as the exposure concept. The definite integral in (3.37) is,
.by definition, the limit of the Riemann sum
R
= L fx+s; · L'.lsi,
n
i=l
(3.41)
where f;+s; represents the number of persons alive at some point in the l1
subinterval of (x,x+ 1). Multiplying by &;, the width of that subinterval,
gives the number of life-years lived in the subinterval. Since fx+s is con­
stantly changing, R is only an approximation to the precise number of life­
years lived in (x, x+ I]. But
lx
=
t f ds = lim R
�
x+s
tt�tt-o
gives the exact value of the exposure concept, where 11..111 is the largest L'.ls;.
By extension,
nLx
=
1"
Px+s ds
(3.36)
gives the exact exposure in the interval (x, x+n]. Further,
(3.39)
clearly gives the exact exposure over the interval (x, oo), which can be
viewed as the total aggregate number of life-years lived in the future by the
gives the average future lifetime per person in the fx
fx group. Then
i"
x
52
r:
The Life Table
group. We have already seen, from (3.30), that
is �x, the expected future
lifetime for a person age x (i.e., in the f\ group). The correspondence of
these two concepts, average and expected, is a very familiar one.
In turn,
(3.32)
also has an exposure interpretation. However, this first requires an under­
standing of the concept and properties of the stationary population, which
we do not wish to presume. The interested reader is referred to Brown [16].
Now that we have the exposure interpretation of Lx , we can give
further meaning to mx, defined by (3.38) as
dx
Lx ·
(3.38)
Recall that in Chapter 2 we defined mx as a continuously weighted average
of the hazard function ..\(y), over the interval (x,x+l]. Equation (3.38) shows
us that, in the life table model, dx persons died in (x, x+ l ], and the associated
quantity of exposure to the risk of dying for the group was Lx. The ratio, mx,
is thus the (annual) rate of death (rate of mortality) in this one-year age
interval. Note how this rate differs from the probability qx , given by (3.4); in
particular, it is possible for mx to exceed one, whereas qx cannot.
For the interval (x, x+n], we have
nmx -
ndx
L.
n x
(3.35)
Note that, although nmx relates to an n-year age interval, it is an average
annual rate, rather than a rate for an n-year period. This can easily be seen
by comparing (3.35) and (3.38). In (3.38), we have a "year's worth" of
deaths in the numerator, and a "year's worth" of exposure in the denomina­
tor. ln (3.35), we have "n years' worth" in both the numerator and denom­
inator, so the relative magnitude of the ratio is still annual.
A modification of this concept of exposure would be to count only
the exposure in (x, x+ 1] contributed by the survivors of that interval. Since
there are fx+l survivors, and each contributes exactly one year, then fx+I is
also the exposure. In similar manner, fx+2 is the exposure in (x+ I,x+2], and
so on. Summing over all years from age x to infinity, we have
T/
=
::,0
Liy .
y=x+I
(3.42)
53
Other Functions Derived from fr
We can describe T/ as the total future lifetime of the fx group, in which only
whole years of future lifetime are counted. Then
T''
(3.43)
-2._
fx
is the average (or expected) number of whole years of future lifetime per
person in the fx group. It is called the curtate expectation of life at age x, as
distinguished from the complete expectation denoted by �xWe can derive an alternate formula for ex as
er =
fx+k = L kPx·
fx · y=x+I
L fy = L · L
k=I
k=I
00
00
00
(3.44)
Finally, if we make the assumption that a person lives approximately one­
half year in the year of death, then we see that �x exceeds ex by that half­
year. Thus
(3.45)
Later in this chapter we will formalize this approximate, intuitive result.
3.3.6 Relationship between nqx and nfflx
Familiarity with the concept of exposure allows us to develop a very useful
formula which relates the functions nqx and nmx. Let us first explore this
and mx =
then we are, in effect,
relationship with n = I. Since qx =
exploring the relationship between fx and Lr. To do this, we first need to
define a new function.
From (3.40), we recall that
t,
f:,
(3.46)
gives the aggregate number of life-years lived in (x, x+ 1] by those who die in
that age interval, namely dx. Then if (3.46) is divided by dx, we obtain the
average number of years lived in (x, x+ l] by those who die in that interval.
It is clear that this average number is necessarily less than one, and could
also be called the average fraction of (x,x+ l] lived through by those who
die in that interval. We define this average fraction to be.fx, so that
The life Table
54
(3.47)
Further, since dx = fx - fx+ ! , then fx+ 1 = fx - dx , SO we have
or
Lx = fx - (l -.fx)dx.
Then
=
(3.48)
(3.49)
Alternatively,
(3.50)
so
_ dx _
dx
=
mx
qx - f - L + (1-fx)d
l + (1-fx)mx ·
x
x
x
(3.51)
Note that, since fx is non-increasing on (x, x+ 1 ], then Lx � fx , so mx ? qx .
More generally, if we consider then-year age interval (.x,x+n], then
(3.46a)
gives the aggregate number of life-years lived in (x, x-:-11] by those who die in
that interval, and there were 11dx such deaths. Division of (3.46a) by ndx
gives the average number of years lived in (x,x+n] by those who die in that
interval, and further division by n, the interval length, gives the average
fraction of the interval lived through by those who die in it. We define this
average fraction to be 11fx, obtaining
I"
nJx
-
_
nl:r -_!.l · f\+n
n · ndx
(3.47a)
Further, Ex+n -- tx -- ndx, so we have
or
nL., -- n · £., -- n( l - nfx )ndx .
(3.48a)
Other Functions Derived from ix
55
Then
(3.49a)
Alternatively,
or
(3.50a)
so
(3.Sla)
Formula (3.5 la) gives the general relationship for deriving nqx from nmx.
This formula is important in the estimation of a life table from population
data. The concept of exposure is also extensively used in that exercise.
!!EXAMPLE 3.71! The following values are taken from a certain life table:
.eo = 100,000
l 1 = 97,408
l5 = 97,015
Lo= 97,764
4L 1 = 388,713.
What are the values offo and 4f1?
!SOLUTION! do = lo -l1 = 2592. From (3.47),fo = Lodo li =. 13734.
4
Similarly, 4d 1 = l1 -ls = 393. From (3.47a), 4Ji = 4 Li4·4d1·ls = .41539.
\!EXAMPLE 3.81! If sqso = . 400, and smso = . I 02, find s/so!SOLUTION! (3 .49a) leads to n · nmx - n · nmx · 11 qx ( 1-nfx) = nqx, so that
(1-nfx) = n·_nmx - nqx. Substituting the given values, along with n = 5, we
n nmx· nqx
obtain (l-5.fso) = .53921, whence sfso = .46079.
0
The Life Table
56
3.4 SUMMARYOFCONCEPTSANDNOTATION
Thus far we have been striving for an understanding of the nature and prop­
erties of survival models, both in the notation of the stochastic model of
Chapter 2 and the life table model of Chapter 3. It might be useful at this
point to summarize the concepts developed thus far, and give the standard
formulas or symbols for them in the notation of both chapters.
TABLE 3.2
Concept
Chapter 2 Formulas Chapter 3 Formulas
1. (Unconditional) probability
of survival from age O to age x
2. (Unconditional) probability
of death not later than age x
3. (Conditional) probability of
survival from age x to age x+n
4. (Conditional) probability of
death before age x+n, given
alive at age x
S(x)
fo
F(x)
S(x+n)
S(x)
fx+n
nPx = ­
l;,
S(x) - S(x+n)
S(x)
5. Hazard rate (force of mortal­
ity) at exact age x
A(x) =
6. (Unconditional) density
function for death at exact age x
f(x) =
7. (Unconditional)
expectation of future
lifetime at birth
xPo =
- f S(x)
= -
S(x)
Jx lnS(x)
d
F(x)
dx
= _ dxd S(x)
'°
µx =
- 1_.ix
fx
= - .i.
dr: Infx
d
xpoµx = - dx xPo
100
G
=
x f(x) dx �o =
E [X]�����
x · xPoµx dx
·1000
To
= .� S(x) dx
=
po dx = 0
Jn"���
�o
���
(
57
Methods for Non-Integral Ages
TABLE 3.2 (Continued)
Concept
Chapter 2 Formulas
8. (Unconditional)
variance of future
lifetime at birth
l'°
9. (Conditional)
density function
for death at agey,
given alive at
agex (y >x)
Var(X)=
Chapter 3 Formulas
=
1
100
00
E[X-xlX>x] = E[SIX>xJ
11. (Conditional)
variance of future
lifetime at agex
Var(SIX>x)
= Var(XIX>x)*
12. Central rate of
death over (x, x+n]
13. Exposure
Concepts
sf(x+slX>x)ds
S(x+s)
ds
o
S(x)
1
00
= s2
nmx
f(x+slX>x)ds
- {E[SIX> x1}
•x
2
10000
1
= 100
�x =
S•
=
0
= 2.
O
2
S
·
sPxµx+s ds
Tx
sPxds = x
f.
sPxµx+sds - ex
o2
f: - ({: ) 2
( no specific symbol)
· .X(x+s)ds
= [:S(x+s)
J:S(x+s)ds
f. +sµ x+s
= J:x
Do not exist in the
Chapter 2 model
nlx = 1 fx+sds
ds
J;ex+sds
d
nLx
nx
= -
n
Tx =
*Variance of future lifetime and variance of age at
death are equal, sinceS = X-x, andx is a constant.
02
Poµxdx - o
e
= yxP- xy
µ
= sPxµx+s,
wheres= y
10. (Conditional)
expectation of
future lifetime
at agex
=
2
( no specific symbol)
2
f(y)
f(ylX> x) =
S(x)
=
0
X
= 2-�-(Tuf
f.o
f.o
x2 f(x)dx
- {E[X]}
100
100
T*
=
x
Yx =
:c
00
fy dy = LLy
00
I:
lx, d
y=+I
ly
Ty y
y=x
58
The Life Table
3.5 METHODS FOR NON-INTEGRAL AGES
A review of the functions that we have developed, and summarized in Table
3.2, shows that not many of them can be numerically determined from a life
table that gives values of fx only for integral x. Actually, only the probability
function nPx (and its complement nqx ) for integral x and n can be so deter­
mined. (Note that xPO and xqo, for integral x, are special cases of nPx and nqx .)
The determination of all other functions requires that values of fx+s
be available for all s, 0 � s � 1. This is obtained in the life table model by
assuming that fx+s has a certain mathematical form between x and x+ 1. This
assumed form for fx+s will be differentiable on the open interval O < s < 1,
but not at s = 0 or s = 1.
The ability to determine fx+s numerically for any s, 0 � s � 1, will
allow us to calculate probabilities of the form sPx and 1-sPx+s , and their com­
plements sqx and 1-sqx+s · The differentiability oUx+s will allow us to evalu­
ate µx+s, and hence the conditional density function f(s IX> x) = sPx µx+s,
for all s on the open interval O < s < I. The integrability of fx+s will allow
us to calculate Lx , Tx and Yx , and hence mx , �x and the conditional variance
of future lifetime (or age at death), Var(XIX> x).
These between-the-integral-ages assumptions are extremely impor­
tant and useful, so we want to acquire a complete familiarity with them. We
will later see that they are essential elements in the estimation of the model,
as well as being useful for making calculations from the life table model.
We should not lose sight of the fact that we assume a mathematical
form for fx+s only between x and x+ 1, not for the entire domain of x; · the
latter case would return us to the continuous parametric models described in
Chapter 2. We will also see that each particular mathematical form that we
assume for fx+s will correspond to a certain interpolation method.
To recapitulate, we assume that we have a life table with numerical
values of f.r given for all integral x. We then assume a mathematical form for
fx+s , 0 � s � l , and show how to calculate, from the given life table, values
of sPx, 1-sPx+s, Lx , Tx , mx and gx · (The functions Yx and Var(XIX> x) can
also be evaluted, but with greater difficulty. Since we will not need them for
our estimation work, we will not develop them in this text.) We will now
pursue each of the following three assumptions of a mathematical form for
fx+s : linear, exponential, and hyperbolic.
Methods for Non-Integral Ages
59
3.5.1 Linear Form/or lx+s
If f.x+s is a linear function between x and x+ I, then it is of the fonn a+bs.
To provide continuity for f.x+s, we require, at s = 0, that f.x = a, and at
S = I, that f.x+ I = a+b, so that b = f.x+ i-a, or b = fx+ 1-f.x = -dx.
Thus we have
(3.52)
An alternate form is to use fx - f.x+I in place of dx , obtaining
(3.52a)
Both (3.52) and (3.52a) reveal that the linear assumption for fx+s allows us
to determine values of fx+s from fx and fx+I by linear interpolation.
The determination of other functions follows easily from (3.52):
Thus
�nd
sPx = ff;s = I - S •
t=
1- S · qx ,
(3.53)
(3.54)
We also have
1-sPx+s
(3.55)
and
l - 1-sPx+s
1-s·qx -Px
1 - S • qx
(1-s). C/x
l - S · qx ·
(3.56)
Next,
1 - S • qx '
(3.57)
which, when multiplied by (3.53), leads to the very convenient result
(3.58)
Note that µx+s, and hence /(s IX> x), are not .defined at s = 0 ands= I,
since fx+s is not differentiable there.
60
The Life Table
Finally we have
(3.59)
so
::,c
Tx = LLy
y=x
Then
mx
and
-
oc
L (ey+1+!dy)
y=x
¥::x = fx - !dx = l - -2I ·qx '
ex = L. fx
0
= T*x + 2,I fx.
dx
T/ + ii\
fx
qx
l
ex+ 2'
(3.60)
(3.61)
(3.62)
which we developed intuitively in (3.45).
Turning to the question of the associated probability distribution
over the interval (x,x+l], we note that the PDF for time at death within the
interval is a constant, and the CDF, which is 5qx , is a linear function. This
shows that the random variable S has a uniform distribution over (x, x-t- I J.
Furthermore, the linear nature of (3.52) shows that fx+s decreases uniformly
(i.e., persons are dying uniformly) over the interval. For these reasons, this
linear assumption for fx+s has traditionally been called the uniform distri­
bution of deaths assumption, which we will refer to as UDO.
Without question the UDO assumption is extremely useful, both for
the purpose of making calculations from the life table, and for the purpose of
estimating the life table in the first place. No doubt the ability to analyze the
assumption in terms of the uniform ( or even) distribution pattern of the
occurring deaths, along with the mathematical simplicity afforded by the
constant PDF, are the major reasons for this assumption's popularity.
Methods for Non-Integral Ages
61
!(tXAMPLE 3.911 Evaluatefx, defined in Section 3.3.6, under the UDD as­
sumption.
[SOLUTION! From(3.47),fx = Lx�:x+I = lx+i+ �x-lx+l = 4,using
(3.59) for Lx. The uniform nature of the time-of-death distribution makes
this result intuitively obvious.
0
3.5.2 Exponential Form/or lx+s
If fx +s is an exponential function between x and x+ 1, then it is of the form
fx+s =a· bs. To assure continuity, at s = 0, we have fx = a, and at s = I,
we have f.x+1 = ab, so that b = lx+a I = �x Thus we have
¥.
(3.63)
which shows that the exponential assumption for fx +s allows us to determine
values of fx+s from values of fx and fx+l by exponential interpolation.
{Note the similarity of (3.63) and (3.52a).) An alternative form of(3.63),
which results from substituting fx+l = fx · Px , is
(3.63a)
Thus
The determination of other functions follows from(3.63) or(3.63a).
(3.64)
sPx
and
(3.65)
We also have
1-sPx+s
and
=
fx+I
s
f(
x Px) -
(3.66)
(3.67)
l - 1-sPx+s - 1 -(px) l-s
Next,
- - lnpx ,
(3.68)
62
The Life Table
a constant, for O < s < 1. Thus we have the useful result that, if fx+s is ex­
ponential, then the force of mortality, µx+s, is constant over (x, x+ I). We
will call this constant force simply µ. Then (3.68), which says µ = -lnpx,
can be rearranged to read
Px
e-µ '
(3.68a)
sPx
e-µs '
(3.64a)
from which (3.64) becomes
and (3.66) becomes
- e -µ(1-s)
1-sPx+s _
(3.66a)
The conditional PDF is then found by multiplying (3.64a) and (3.68), obtain­
ing
(3.69)
f(s IX> x) = sPx µx+s = µ · e-µs.
The PDF of S, given by (3.69), and its SDF, given by (3.64a), clearly
show us that S has an exponential distribution over (x, x+ I]. (Recall from
Section 2.3.2 that this distribution has a constant hazard rate, which is the
same as the force of mortality.) This constant force of mortality has led to
this assumption being traditionally called the constantforce assumption.
We will find that the exponential assumption is not as convenient as
the linear assumption for making calculations from the life table, but, on the
other hand, it is sometimes more convenient than the linear for the purpose
of model estimation. We will make frequent use of both assumptions in Part
II of this text.
With respect to the exposure functions Lx and Tx , we find
Then the central rate is found to be
-dx. = �
- µ,
µ(3. 70)
d
(3.71)
? = µ,.
dxµ
This result comes as no surprise, since, from (2.59), mx is the weighted
average value of µx+s· If µx+s is constant, then clearly its weighted average
is the same as that constant, which is µ.
mx = !}:x =
Methods for Non-Integral Ages
63
Of course thisµ is the constant value ofµx+s, 0 < s < 1. For clarity
let us call it µx . For the interval (x+ I, x+ 2) we have a different constant
force, namelyµx+ 1 = - In Px+ 1 , so Lx+ 1 = dx+ 1 /µx+ I· Then we have
(3.72)
There is no convenient way to further simplify(3.72), but it does show that
Tx can be precisely calculated from a life table under the exponential as­
sumption.
Finally, �x can be obtained as Tx /fx . Again there is no convenient,
simple formula for �x!!EXAMPLE 3.10� Evaluate fx, as defined in Section 3.3.6, in terms of Px
under the constant force assumption.
{SOLUTION! Weknow thatlx = � = -l�� · Thenfrom(3.47) we have
x
)
ix -= (dx l-lnpdxx - fx+ 1
=
=
l
f.x+I
- lnpx - dx
1
&. =
- lnpx - qx -
1
__h_
- lnpx - 1 - Px ·
D
3.5.3 Hyperbolic Formfor lx+s
t,
If fx+s is a hyperbo lie function between x and x+ I, then it is of the form
1
fx+s = (a+bs)- • To assure continuity, at s = 0, we have fx = �, or a=
I
I
I
I
- - a = -'-x+I
o.
Th us
and at s = I, we have J\+ 1 = a+b, so that b = -.'-x+I
. - - <-x
or
fx+s =
(If
x
+S
{ f 1+ l - f1})- 1 '
x
x
I
l
= l
( I
fx+s - fx + S fx+I - fx
-1- + (l-s) · .l,
= s · fx+I
e.T
(3.73)
)
(3.73a)
64
The Life Table
I can be determined by linear interpolation
which shows that values of e
x+s
between the reciprocals of fx and fx+ l · Linear interpolation on the reciprocal
of a function is called harmonic interpolation on the function itself.
From (3.73) we can determine expressions for other functions. To
obtain sPx, we consider first
(sPx)
-1
fx
= t
x+s
so that
sPx
Then
We also have
=
(3.74)
sqx = l - sPx
(3.75)
(1-
-1-})
1
l'x+I .fx +s{l'x+ I
fx
= Px + s(l-px) = l - (l-s)qx,
1-sPx+s
(3.76)
from which we find the very convenient result
1-sqx+s = 1 - 1-sPx+s = (1-s)qx.
Next, using sP x =
=
/J,x+s -
d
(3.77)
Px
, we find
Px +s ·qx
- cfs sPx
sPx
=
=
Px · qx
Px
•
2 (px +S • qx ) • Px + S · qx
qx
qx
_
Px + S · qx - I - (l -s)qx '
(3.78)
and so
(3.79)
Again we note that µx+s and/(s IX> x) are not defined at s = 0 ands= I.
Methods for Non-Integral Ages
65
The hyperbolic assumption is somewhat illogical, in that it implies a
decreasing µx+s over (x, x+ 1), whereas the linear assumption implies an
increasing µx+s, and the exponential assumption implies a constant µx+s, as
we have seen. It is not often used for making calculations from the life table
model, and is not widely used in estimation work outside of actuarial sci­
.ence. For these reasons it might be thought that this assumption need not be
considered any further. However, the convenient relationship
(3.77)
is used in what has been the traditional actuarial approach to survival model
estimation for over a century. This approach is not without merit, and will
no doubt continue to be used in actuarial circles, so the hyperbolic assump­
tion will continue to receive some attention.
The Italian actuary Gaetano Balducci made major use of the hyper­
bolic distribution in several of his writings, such as [5] and [6]. Although he
did not originate the use of this assumption, it has come to be called the
Balducci assumption, or Balducci distribution.
With respect to the exposure functions Lx and Tx , we note first that,
I
I
I
I
, where a=-.
and b = <-x+I
from (3.73), fx+s = Then
o - - T.
a +bS
'-x
<.x
Lx =
1'
fx+s ds =
1'
a t\s =
i · In (a; b).
Substituting for a and b, we find a! b = �x and t =
!.;:'. Thus we have
L _ fx+ I . 1 J_ _ _ o ( In Px) .
x qx n Px {.x+I qx
(3.80)
From Lx we can find Tx , mx and ex from the usual formulas.
0
!!EXAMPLE 3.111! In a certain life table, fx = 1000 and fx+ l = 900. Eval­
uate mx under each of the UDO, constant force, and Balducci assumptions.
!SOLUTION! From the given values we find dx = 100, lnpx = -. I 0536, and
µ = - lnpx = .10536. Under UDO, Lx = fx - idx = 950, so we obtain
mx = = .10526; under constant force, we directly have mx = µ = .10536;
- fx�; lnpx = 948.24, so m = �� � = .10545.
under Balducci, Lx =
x
9 - 4
t
D
66
The Life Table
3.5.4 Summary
Table 3.3 summarizes most of the results developed in this section. Our pur
pose is to establish a sufficient familiarity with the three common assump­
tions to allow an understanding oftheir use in survival model estimation. A
deeper analysis of these assumptions is given by Batten [8], and Mereu [ 55]
gives a presentation ofthe use ofthese assumptions in actuarial calculations.
Function
Cx+.,·
Linear
(UDD)
l!x-s·d,
= S · i\+1
+ (1-s)·f!x
sPx
1 - S • qx
s qx
S • qx
1-sPx+s
1 -s · q;x
Px
TABLE3.3
Exponential
(Constant Force)
l )-I
(
I
fAPxY
+(1-s)· fx
£x
·
i
l -s
s
+
= Ux+lY · (£x)
px
Px + s(l-px)
1-q..
- l - (1-s)qx
S • qx
s
(px)'== e-µ(l-s)
sPxflx+s
Lx
fx - !dx
= f I + -}d
Hyperbolic
(Balducci)
dx
;x
µ
---- --·-x+
�. ..'.-- _ --=-i-----------
1 ·- (1- s)qx
-
Px + s(I-Px)
= I - (1-s)qx
67
Methods for Non-Integral Ages
3.6 SELECT LIFE TABLES
In this section we consider life tables developed from the select survival dis­
tribution defined in Section 1.2 as S(t;x), t � 0.
It is easy to see that a life table based on this SDF is completely
parallel to one based on S(x). Thus we begin with a radix which represents a
hypothetical cohort of lives selected at age x, which we call f[x] . Then subse­
quent values are developed from
f[x]+t = f[x] • S(t;x).
(3.81)
If values of f[x]+t are specified for all integral t, the resulting select
life table (for age at selection x) can then be used to obtain other values, just
as we did for the aggregate life table earlier in this chapter. It is important to
specify the age at selection in the symbols for the functions derived from a
select table. For example, if selection is at age x, the conditional probability
of death between ages x+ 7 and x+ 10, for a person known to be alive at age
.x+5, is given by
_ f[x]+7 - f[x]+IO
213 q[x]+5 f[x]+S
(3.82)
Select life tables do not normally extend to all values oft in a man­
ner that is unique to the selection age. For example, consider two persons,
one selected at age 20 and now age 30 (t = l 0), and the other selected at age
21 and now age 30 (t = 9). Their conditional probabilities of survival for
·
, respect1veanother n years are nP[20J+10 = fr201+10+n
and nP(21]+9 = f[21]+9+n
•
e
�1201+10
(21]+9
ly. If it is felt that selection no longer has an effect on mortality at these
durations, then the two probabilities should have the same value, denoted by
l3o+n A life table with this characteristic is called a select and ultimate
l30
table.
The first step in developing a select and ultimate table is to deter­
mine the length of the select period, the time over which selection is
assumed to have an effect on mortality. Although there is evidence that
effects of selection can persist for many years, published tables will seldom
show a select period greater than 15 years. For illustration in this text, how­
ever, we will assume only a four-year select period. This means that for
persons four and more years beyond selection, mortality is a function of
attained age only.
68
The Life Table
Suppose [x] is the youngest age at selection in the table which we
wish to construct. Then we select a radix, l'[x], and generate
and
l'[x) · S(t;x) fort= 1,2,3
(3.83a)
l'[x] · S(t;x) fort � 4 .
(3.83b)
Instead of arranging these values of l'rxJ+i and fx+t in a column, as in our
earlier Table 3. I a, we arrange them in the row-and-column format shown in
Table 3.4 .
TABLE3.4
f[x]+2
f[x]+3
Next, consider age at selection [x+ l]. If a radix, Rrx+lJ, were arbitrar­
ily chosen, then we would generate
and
frx+ l ]+1 - f[x+ I J • S(t;x+ 1) for t = 1,2,3
(3.84a)
£.,+ 1 +1 - f[x+ lJ · S(t;x+ l) for t � 4.
(3.84b)
But the basic idea of the select and ultimate table is that the same values of
fx+t, t � 5, should be common to the cohorts CrxJ and frx+1J· Thus frx+IJ must
be chosen so that frx+ lJ · S(4;x+ l) produces the same f..t+s as was produced
by £[xJ · S(S;x). This is easily accomplished by taking
fx+5
.e[x+I] = S(4;x+
l )"
(3.85)
Then frx+IJ+l, Prx+IJ+2, and frx+IJ+3 can be found from this fir+iJ by using
(3.84a). Note that values of S(t;x+ 1) fort> 4 are not utilized once it has
been decided to use a four-year select period.
Similarly, l'rx+2J would be found from the already established value
+-6
by
of fx
Methods for Non-Integral Ages
69
l\+6
= S(4;x+2)'
(3.86)
and then values of f'[x+2J+r, t = 1,2,3, are found from
frx+2J+1 = frx+2J · S(t;x+2).
(3.87)
The complete select and ultimate table is in the format of Table 3.5,
where we assume age 20 is the youngest age at selection.
TABLE3.5
[x]
--
f'[x]
f'[x]+I
f'[x]+2
icx]+3
fx+4
x+4
[20]
ir201
£(20]+1
£(20)+2
Rr201+3
£24
24
[21]
£(21]
£[21)+1
fr211+2
£(21)+3
i2s
25
[22]
l'r221
l'[22J+l
l'r221+2
l'[22]+3
£26
26
From this table, conditional probabilities can be calculated for integral ages
and durations. Other calculations can be made by assuming one of the mor­
tality distribution assumptions, as described in Section 3.5, between pairs of
adjacent integers.
!!EXAMPLE 3.121! Express the conditional probabilities (a)2,4q[2oJ+I and
(b)214q r221+3 in terms of f's, assuming a 4-year select period.
!SOLUTION[
(a) This is the probability of death between ages 23 and 27 for a person age
21, selected at age 20. The number of such deaths, from Table 3.5, is
. £120)+3 -(27
. d probab1"l"1ty 1s
{,{)(20]+3 - {,{)27, so the des1re
n
<C(20]+1
(b) Here we seek the probability of death between ages 27 and 31 for a
person age 25, selected at age 22. For this person, age 27 is already
O
beyond the select period, so the probability is lf.7 - l31
[22)+3
In this section we have developed the select and ultimate life table
from values of S(t;x). Alternatively, a set of q[x]+r, t = 0, 1,2,3, for all ages
70
The Life Table
at selection, and qx+t, t 2: 4, might be available. Since P[x]+t = 1 - q(x]+t
and Px+i = 1 - qx+t , then values of S(t;x) can easily be found. Thus the
development of a select life table from a set of q's is totally parallel to its
development from S(t;x). For a description of this select table development
from q's, specifically oriented to the idea of selection for issuance of insur­
ance, see Section 1.8 of Jordan (41].
3.7 SUMMARY
The life table is a very common form in which survival models are ex­
pressed, expecially in actuarial and demographic work. In this chapter we
have tried to provide a very thorough description of this model, including an
understanding of the correspondence between the life table and the concep­
tual survival distribution expressed by S(x) and its related functions.
Detailed uses of the life table is not the subject of this text. We have
described tabular models in this chapter as background for the material on
estimating such models, which comprises Part II of the text. In particular,
the material in Section 3.5 on mortality distribution assumptions will be
useful in the estimation of models.
Some explanation would be useful at this point concerning our nota­
tion (x, x+ l] for the interval from x to x+ 1. We take the view that we are
addressing the question of death (or survival) over this interval, given alive
at x. If the given condition is alive at x, then it follows that death at x is not
possible for this interval. Death at x+ l is possible for this interval, however.
In other words, death at an integral age, which is an interval boundary, is
defined to "belong" to the interval closing at that boundary, not the one
opening there.
It must be understood that this choice was somewhat arbitrary, and
other texts might choose the opposite, thereby using [x,x+ 1). Our choice is
based on a literal interpretation of the condition alive at x, as well as a desire
to be consistent with the unconditional probability F(x) = Pr(X::::; x).
The reader should also be aware that when one calculates the conditional probability qx =
the issue of whether this is the probability of
death prior to x+ 1, or not later than x+ I is not numerically crucial, since we
may take .Pr(X = x+ 11 X > x) to be zero. But for the estimation work in
Part II of the text, if a death is observed at age x, then, as a practical matter,
we must decide whether to consider it as being "between x-1 and x" or
"between x and x+ l." We will choose the former, and this is indicated by
the notation (x-1,x] and (x,x+ 1).
t,
Exercises
71
3.8 EXERCISES
3.1 Introduction
3.2 The Traditional Form of the Life Table
3-1.
A tabular survival model is defined by the following values ofPx :
x Px
0 .9
l .8
2 .6
3 .3
4 0
(a) What are the corresponding values of S(x), for x = O,J,2,3,4,5?
(b) Using a radix of 10,000, derive a life table showing the values of
f:x and dx.
What is the value of w in this table?
c
)
(
w-1
( d) Verify that L dx = fo.
x=O
3-2.
From the life table developed in Exercise 3-1, calculate each of the
following:
(b)iq1
(d) 3q2
(c) 3p1
(a) 3do
3-3.
The unconditional probability of death between ages x and x+ 1 is
given by xlqo.
( a) Define x )qo in terms of S(x) and in terms of fx .
w-l
(b) Show that L xlqo = l.
x=O
3-4.
A survival distribution function is defined by S(x) + ��;, 0 s;: x s;: c.
A life table is developed from this SDF using £0 = 100,000. In the
resulting life table, £35 = 44,000.
(a) Find the value of win the table.
( b) Find the probability of surviving from birth to age 60.
(c) Find the probability that a life age IO will die between age 30
and age 45.
72
The Lzfe Table
3.3 Other Functions Derived from lx
3.4 Summary of Concepts and Notation
l
wl-
3-5.
If the force of mortality is µx = x i +
x, 0 :::; x < I 00, find
the number of deaths which occur between ages l and 4 in a life table
with a radix of I 0,000.
3-6.
Find each of the following derivatives:
a
(a) os sPx
a
(b) ox sPx
3-7.
3-8.
l
l
A simple approximation to :Jx fx is x+i; x-i Use this approxima­
tion to determine a value for µ2 from the life table shown in Table
3. la.
If fx = 2500(64-.8x) 113 , 0 ::; x::; 80, find each of the following:
(a) f(x)
(b) E[XJ
(c) Var(X)
3-9.
Find each of the following derivatives:
(a) !l. U:, lj
dx
(b)
Le,,
J� (ln Tx)
3-10. For the fx function given in Exercise 3-8, find �7o and the variance of
future lifetime for a person age 70.
3-11. Showthatlx+I = Lx · exp [ - fx
x+I
mydy]. (cf, Equation (3.10))
3-12. Evaluate 10m10 from the fx function given in Exercise 3-8.
Exercises
73
3-13. Find each of the following derivatives:
a
(a) Bx nLx
a
(b) Bn nLx
3-14. Consider the fx value in a life table to represent a hypothetical group
of persons all age x. Give symbolic answers to each of the following:
(a) What is the aggregate number of life-years that this group will
live in the future (their total future lifetime)?
(b) How many life-years will they live prior to reaching age x+n?
(c) What is the average (or expected) lifetime for a person in this
group prior to age x+n (denoted by �x:n1)?
(d) How many in this group survive to age x+n?
(e) How many life-years did the survivors in (d) live from age x to
agex+n?
(f) Then how many years were lived between the ages of x and x+n
by the persons who died in that age interval?
(g) How many persons died between ages x and x+n?
(h) What was the average number of life-years lived between ages x
andx+n, by those who died in that age interval?
3-15. Show that ex = PxO + ex+1).
3-16. Let e x be the curtate expectation of life at age x according to a certain
life table, and let e: be the same concept according to a 'second life
table. For all x it is known that 11 !;x = I +k, where k is a.-.constant.
Show that q: = qx + 1 +�x+t.
x
3-17. Let µf denote the force of mortality in Life Table I, and µfl denote
the force of mortality in Life Table II. If µf1 = µf + ;b, show that
e,
74
The Life Table
3-18. Consider the function ix defined by ( 3.47). Which of the following
correctly define ix?
1
S • sPxµx+s ds
(a)
J�
(b) E[X-xjx:SX:Sx+l]
(c)
.C+ 1 (y-x) · S(y) · .\(y)dy
x
S(x)-S(x+l)
3-19. Determine sho from the following data:
x
70
75
80
85
fx
54,900
40,900
26,100
13,700
Tx
573,800
332,800
165,800
67,600
3-:20. Let nax be the average number of years lived beyond age x by persons
who die between ages x and x+n. Given that £35 = 1060, £40 = 960,
5m35 = . 020,and 5m40 = . 018, find 5a35.
3.5 Methods for Non-Integral Ages
3-21. Show that linear interpolation on the log of a function is the same as
exponential interpolation on the function itself.
3-22. If tx = lOOOJioo - x, 0 :S x :S 100, calculate the exact value of
µ36+1;4, and compare it with the value obtained from each of the lin­
ear, exponential and hyperbolic assumptions. (Round the value of £37
to the nearest integer.)
3-23. Iffx = 15,120andqx = t,findfx+1;4under the hyperbolic assump­
tion.
75
Exercises
3-24. The following values are given by a certain life table:
10,000
Lx+ J
= 8,000
.2500
8,100
Lx+2
= 6,000
mx+ 2 - .3645
Using exponential interpolation, find 2qx+l/2·
3-25. Let µx +k denote the constant force of mortality for the age interval
(x+k, x+k+1). Determine �x:JI, the expected number of years to be
lived over the next three years by a life age x, given the following
data:
k
0
I
2
exp( -'- µx+k)
1 - exe.{ -1!:x+k)
µx+k
.9754
.9744
.9730
.9512
.9493
.9465
3-26. If 1 t/!+1;2 = zq;+1;2 = k, find z in terms of k (Hand L refer to the
hyperbolic and linear assumptions for fx+s, respectively.)
3-27. Which of the following relationships are correct under the linear
assumption for fx+s , 0 < s < I?
(a) 1/2qx < 1/2qx+I/2
(b) sqx = 1-sPx · sqx+ t-s
(c) µx+s > sqx
3-28. Show that Yx = I)k+ I) · fx+k+ 1 +
k=O
! · R , under the uniform distribu­
x
tion assumption, where x is an integer.
3-29. Given the following excerpt from a life table, find the median of fu­
ture lifetime for a person age 50, assuming (a) uniform, and (b) con­
stant force.
fso = 80,000
£74 = 42,693
f1s = 40,280
f.16 = 37,480
76
The Life Table
3.6 Select Life Tables
3-30. For a select and ultimate table with a four-year select period, show
0
0
0
that e[2oJ = e[lOJ :4! + 4P[20J · e24.
3-31. A select and ultimate table with a three-year select period begins at
selection age 0. Given the following, find the radix f[OJ·
f6 = 90,000
- I
q(OJ 6
d.-r: = 5,000 for x 2: 3
3P[OJ+I =
sPpJ =
(o · 3P[IJ
!
PART II
ESTIMATION OF SURVIVAL MODELS
FROM SAMPLE DATA
This part of the text deals with the mathematics of various approaches
estimating a survival model from sample data.
Once a sample is selected, a basic distinction is made between the case
where the sample is kept under observation until all have died (complete
data sample), versus the case where units in the sample may terminate
observation while still alive (incomplete data sample). Estimation from com­
plete data samples is considerably more straightforward than estimation from
incomplete data samples. The former case is considered in Chapter 4, and
the latter in Chapters 5 through 8.
Chapter 5 is devoted to an analysis of study design to discover the types of
estimation problems that we will encounter.
Chapter 6 describes moment estimation procedures, including the traditional
actuarial approach.
Chapter 7 deals with maximum likelihood estimation, including the product
limit estimator.
Chapter 8 deals with the estimation of parametric survival models, from both
eomplete and incomplete data samples. It includes the topic of hypothesis
testing of such estimated models, and a brief discussion of allowing for con­
comitant variables as well.
In all cases, an attempt is made to analyze the nature and quality of the resul­
ting estimates.
CHAPTER4
TABULAR SURVIVAL MODELS ESTIMATED
FROM COMPLETE DATA SAMPLES
4.1 INTRODUCTION
We begin our study of the estimation of survival models with a very simple
study design. This design is found in clinical studies where the observer can
keep each study unit under observation for the fairly short time until it fails,
and no units are permitted to withdraw from the study without failing (dy­
ing). This situation is referred to as complete data. In those situations where
study units can withdraw, or are otherwise lost from observation before
death, some information regarding the survival pattern of the entities under
study is lost, and we say that the data are incomplete.
To say that the presence of withdrawals complicates the estimation
problem is a huge understatement. We will see that survival model estima­
tion from complete data and the analysis of such estimates are considerably
more straightforward than will be the case with incomplete data.
Because complete data situations are generally found only in clinical
studies, we use S(t), rather than S(x), for the operative survival function
which we wish to estimate. The variable of interest is, therefore, time of sur­
vival (hence failure} since some initial event. Our estimate of S(t) win be denoted by S(t) in most cases.
4.2 STUDY DESIGN
In this chapter we are dealing with the longitudinal study design defined in
Section 1.5. All study units come under observation at a well-defined time
t = 0, and are observed over time until all have died; the exact time of each
death is recorded. The initial group of study units forms a cohort group. By
this we mean a closed group which allows no entrants after time t = 0.
The cohort could be gathered in either of two ways. Suppose, for
example, that 100 laboratory mice are injected with a dose of nicotine on a
certain day. Then time t = 0 is obtained on the same day for all study units,
and, furthermore, a death observed at duration t would occur on the ith day of
the study. In this case it is easy to see the idea of survival viewed as time
since the initial event.
80
Tabular Survival Models Estimated from Complete Data Samples
Contrast this with a second example. Suppose a study is made of the
survival pattern of persons diagnosed as having a certain terminal disease.
The study includes all persons so diagnosed during calendar years 1990 and
1991, and each person has his or her own date on which t = 0. Together the
dates of diagnosis and of death give the observed value oft at death, and this
is the variable of interest for our study. Note that the chronological ages of
the persons are not of concern to us.
!!EXAMPLE 4.1![ Person A was diagnosed on March I, 1990 and died on
March 11, 1991. Person B was diagnosed on July 1, 1991 and died on Aug­
ust 1, 1991. What are their observed times of death, measured in days?
!SOLUTION! Person A died at t = 375 and Person B died at t = 31. It is a
simple matter, but a very important one, to distinguish between calendar
time (or cltronological time), and time under observation (frequently called
follow-up time). A time diagram of the contributions of Persons A and B to
our study, in terms of calendar time, is shown in Figure 4.1a.
A��_............--...-.-._.-.--.....��������
B ���������-�����---�--------��
l/1/92
1/1/91
1/1/90
!FIGURE 4.la!
In terms of follow-up time, the contributions of Persons A and B are shown
in Figure 4.1 b.
A�����--�--�-����B ---
0
31
!FIGURE 4.lb!
375
The idea that we wish to reinforce is that all persons come under observation
at t = O, even though these generally occur at different dates for different
persons. It should also be understood that our procedures for estimating the
survival model will not depend on whether each t = 0 occurs on different
0
days or on the same day.
Exact Time of Death
81
4.3 EXACT TIME OF DEATH
Suppose a cohort of 5 study units is observed, with deaths occurring at or­
dered times ti, 12, t3, !4, and /5. Based on this sample data we wish to
estimate the survival distribution applicable to these entities.
4.3.1 The Empirical Survival Distribution
A very natural approach would be to estimate the operative S(t) by the ob­
served proportion of the sample surviving to each time point t. A graph of
this observed survival distribution (or empirical survival distribution)
would be helpful, and is shown in Figure 4.2. We will use S0 (t) to denote
this distribution.
Values of S0 (t), which estimate S(t), are easy to read from the grayh.
For all t < /1, the proportion surviving to that t is 1.00, and we take this as
our estimate of S(t). We are not saying that death prior to time t1 is impos­
sible, but rather that S0 (t) = 1.00 is the estimate of S(t) which results from
this data and this estimation procedure.
Similarly, we take our estimate of S(t 1 ), which we call S0 (t1 ), to be
.80, since this is the proportion which we observe surviving to time t 1 • Actu­
ally, we take S0(t) = . 80 for all t such that t 1 � t < t2, since .80 is the
surviving proportion at all such t. In like manner we take S0 (t) = .60 for all t
such that t2 � t < /3.
What should be our estimate of S(t) for all t � t5 ? Since our estima­
tion procedure has been defined to be simply the observed proportion survi­
ving, it follows that we take S0(t) = 0 for all such t. Here we are not saying
that survival beyond t5 is impossible, but rather that S0 (t) = 0 is the estimate
of S(t) from this data, according to this procedure.
l.00 ..------...
.80
S0 (t): the observed proportion surviving to time t
.60
.40
.20
!FIGURE 4.2!
Tabular Survival Models Estimatedfrom Complete Data Samples
82
To summarize, for an initial cohort of size n, this simple estimation approach
produces the estimate
l.00
s (t) = { :;;- i
o
for t < /1
for t1 s; t < ft+ 1,
for t 2: t,,
l
= 1,2,- · ·, n-1.
(4.J)
To some extent the constancy of S°(t) for all t between t; and lz+i is
not intuitively appealing. We would expect S(t) to be a strictly decreasing
function oft, and thus our estimate S 0 (t) should show this property as well.
We recognize that the "step-down" nature of S0 (t) results from the finite
sample size from which it was produced. Suppose that, for the entities under
study here, very few ever survive beyond time t = 10. Suppose also that we
have a cohort of n = 50 study units, rather than only 5. Then the empirical
proportion surviving graph would step down .02 each death, and would pack
50 such steps between t = 0 and t = 10. The graph would look something
like Figure 4.3.
S0 (t)
1.00
.90
.80
.70
.60
.50
.40
.30
.20
.10
0
t
2
4
6
8
10
!FIGURE 4.3!
The greater number of steps, resulting from the greater sample size, packed
into a maximum practical range (here taken to be 10), results in estimated
values of S(t) that are "smoother" than those produced from the smaller
sample. If we wish, we could revise the S0 (t) shown in Figure 4.3 by super·
imposing a smooth curve which closely fits the observed step pattern. This
83
£xacl Time ofDeath
revision is called graduation, and would be done as part of the process of
producing an estimate of S(t) for practical use. In this text we are concerned
only with procedures which produce the initial estimate of S(t), and will not
pursue the issue of graduation. (For a thorough treatment of graduation, see
London [52].)
Note that in the estimation procedure described here we assume that
of
death
is recorded with sufficient accuracy such that there is no more
e
tim
than one death at each time point. If the recording unit (i.e., week, day, etc.)
is not sufficiently fine, or if the sample is sufficiently large, or both, then
multiple deaths will occur at one time point. When this occurs, our step
graph of S0 (t) will merely step down � if there are two deaths at that point,
or, in general, Ii for r deaths at the same time point. This is illustrated in
Example 4.2 and Figure 4.4.
[EXAMPLE 4.2� A cohort of eight individuals is observed from time t =O
until all have failed. Only the day of failure is recorded. (By assuming that
the initial event defining t = 0 occurs in the middle of the day, and failure
also occurs in the middle of the day, then all durations at failure are inte­
gral.) The observed durations at failure were 3, 4, 5, 5, 7, 10, 10, 12. The
observed proportion of the sample surviving, S0 (t), is used to estimate the
operative (but unknown) S(t). (a) What are our estimates of S(2), S(5), and
S(l2)? (b) Graph S0 (t).
!SOLUTION!
(a) S°(2) = 1.00, the proportion still surviving. Since we have defined S 0 (t)
to be right continuous, by (4.1 ), the observed proportion surviving at
t = 5 is S0 (5) = .50. Similarly, since 18 = 12 is the time of the last death,
then S0 (12) = 0.
(b)
so (/)
1.000
.875
.750
.625
.500
.375
.250
.125
-.-.
I
I
I
·----c
I
0
3
4
5
7
!FIGURE 4.4j
10
12
t
0
84
Tabular Survival Models Estimatedfrom Complete Data Samples
In general, we might use the procedure of this section for cohorts
small enough to avoid a prohibitive amount of computation. For larger sam.
ples, we can use the approach described in Section 4.4.
4.3.2 Analysis of the Empirical Survival Distribution
We note that S"(t) is an estimate of S(t), such estimate depending on the
random occurrence of deaths prior to (and hence following) time t. For any
fixed value oft, we define S0 (t) as
"
t =
S ()
number ofsurvivors at time t
n
= N
1
n'
(4.2)
where N, denotes the (random) number of survivors at time t. (It will be our
convention to use capital letters for random variables.) Equation (4.2) is the
estimator for S(t). Recall that S(t) represents the "real," or operative, prob­
ability of survival to time t for an entity in our study. Then N, is a binomial
random variable with parameters n (sample size) and S(t), if the standard
characteristics of a binomial model are assumed to hold. We recall that
= n · S(t)
(4.3a)
= n · S(t) · F(t),
(4.3b)
E[N,]
and
Var(N,)
since F(t) = I - S(t).
Since N, is a binomial random variable, then S"(t) = N, is a binop
mial roportion random variable, with
(4.4a)
and
(
Var[S"(t)] = -\ · Var(N1 ) = S t) ;/Ct)_
n
(4.4b)
Equation (4.4a) shows that S0(t) is an unbiased estimator of S(t).
Of course S(t) is not known, so if we want to have numerical values
of Var [S0 (t)] the best we can obtain are approximate, or estimated, values by
using S0 (t) in place of S(t). Thus we have
Var (So(t)] =
whereF0 (t)
l - So(t).
°
0
S
- (t) · F (t)
n
(4.5)
Exact Time ofDeath
85
[iXAMPLE 4.311 Suppose it is known that the survival distribution to
which our sample in Example 4.2 is actually subject is uniform on (0, 15]. (a)
What is the variance of the random variable S0(6)? (b) If the uniform
distribution assumption is not made, how would we estimate Var [S0 (6)J?
1£0LUTIONI
6 4
(a) If Tis uniform on (0,15], then S(6) = .60 and Var[S0 (6)] = <- �- ) = .03.
5 5
(b) Note that S0 (6) = .50, so (4.5) gives est. Var [S0 (6)] = <- �· > = .03125.
[Note: The data of this sample could be used to test the hypothesis that
the applicable distribution is uniform on (0,15). We will do this in
D
Chapter-8.]
4.4 GROUPED TIMES OF DEATH
If the cohort group under observation is large, so that the procedure of the
preceding section is not practical, we might use a grouped deaths approach.
We begin by dividing the time continuum into an indefinite number of fixed
intervals of equal length. This time line is shown in Figure 4.5.
0
1
2
-1t--1- - - - - I
t+l
k-1
!FIGURE 4.51
Although intervals of equal length are not required for this procedure, it is
convenient to assume them. The notation and formulas for the estimators are
simplified by the use of equal length intervals.
4.4.1 Notation
As before, let n be the initial size of the cohort under study. Since we are
dealing with complete data, all n study units are observed to die, and each of
the n deaths falls into one of k intervals, where k is set large enough so that
survival to time k is impossible.
Let D, be the random variable for the number of deaths which occur
in the (t+IY' interval (i.e., between t and t+l ), 0 :$ t :$ k-1, and let d,
represent the realized value of D1 in an actual study. We note that
Tabular Survival Models Estimated from Complete Data Samples
86
n
(4.6)
Let N1 be the random variable for the number of survivors at time t ,
and let n1 represent the realized value of N1 in an actual study. Note that,
necessarily, No = n, and Nk = 0, so No and Nk are degenerate random vari­
ables. We note the relationship
N1+1
= N1
-
D1 , fort = 0,1,- · ·, k- I.
(4.7)
Let q1 represent the conditional probability of death between t and
t+ 1, given alive at t, and let p1 = l-q1 be the complementary survival prob­
ability.
Since S(t) is already understood to be the unconditional probability
of survival to time t (from time 0), then S(t) · q1 is the unconditional proba­
bility of death between t and t+1. In standard actuarial notation we would
use 1lqo for S(t) · q,.
In an actual study using a cohort, complete data, grouped deaths de­
sign, our objective will be to estimate the operative S(t), along with some
other functions of the survival model, and to analyze the estimates we obtain.
To do this it will be important to have a clear understanding of the basic
random variables in our study, since our estimators will depend on those ran­
dom variables.
4.4.2 The Distributions ofN, and D,
Each M, the number of survivors at time t, is a binomial random variable
with parameters n and S(t). The mean and variance of each N1 are given by
(4.3a) and (4 .3b), respectively.
Together the set of random variables D0 ,D 1 , • . • ,Dk-t has a joint
multinomial distribution. The multinomial probability for D1 is S(t) · q1 = 1 lq0.
We recall the following properties of the multinomial distribution:
Pn(do, di,··, dk-1)
,
k-1
.
= d0·rdI····
TI (1 1 qol
,n. dk-1·I t=O
I
1
(4.8)
(4.9)
(4.10)
Grouped Times ofDeath
87
(4. li)
Note that for this distribution, the mean, variance, and covariance are all un­
conditional, i.e., given only n study units existing at time t = 0.
Conditional on N, = n, survivors at time t, D1 is a binomial random
variable, with mean n1 · q, and variance n1 • q1 • p1.
4.4.3 Estimation ofS(t) and ,lqo
It should be clear that S(t) can be estimated by using the observed proportion
surviving tot. Thus
S(t)
N,
= n
(4.12)
is a binomial proportion random variable, with mean and variance given by
(4.4a) and (4.4b), respectively, withS0 (t) replaced byS(t).
Similarly, Mo is naturally estimated by using the observed relative
frequency of deaths between t and t+ 1. Thus
,lq0 = �', t = 0,1,... ,k-1
(4.13)
is a multinomial proportion, with mean and variance given by
E[ ,I qo] = � · E[D,] = rl qo
(4.14)
and
(4.15)
We observe from (4.14) that the multinomial proportion estimator ,lq0 = �,
is unbiased, as defined in Appendix A.
4.4.4 Estimation of q, andPt
Recall that q, is the conditional probability of death not later than time t+ I,
given alive at time t. It is natural to estimate this probability by
"
D,
q, - n,.
(4.16)
88
Tabular Survival Models Estimatedfrom Complete Data Samples
Here we recognize that, conditional on there being n1 survivors at time t, D1
. b.monua
. 1 , so q,
/\ = D,
is
is binomial proportion, with conditional mean and
n,
variance given by
E[q,I ni]
= nl, · E [D1] = q,
(4.17)
and
l
Var(D1 )
= l - q,
n,-D1
n1
Var(q1 I n1 )
Furthermore,
/\
P1
n2
1.v(l-q1)
n1
(4.18)
N,+1
= n;-
(4.19)
is also a binomial proportion, so
E[P1 \ n,] - Pr
and
Var(p1 lnt)
= p,(��p,).
(4.20)
(4.21)
Since q1 = 1-p,, then, clearly, (4. 18) and (4.21) are the same.
11.@MPLE 4.411 Consider a sample of 20 individuals existing at t = O.
All fail within 5 weeks, and only the week of failure is recorded. The ob�
served outcome of this sample is that two individuals failed in the first week,
three in the second, eight in the third, six in the fourth, and one in the fifth.
Using the data of this sample, estimate (a) 2lq0, (b) S(3), and (c) q3.
lfil)LUTION:] Figure 4.6 is a diagram of this study.
20
��-��2=--1 �-�3�,t--��8�,t----�-6"--r-l ���
1 -, 0
1
2
3
4
5
!FIGURE 4.61
89
Grouped Times ofDeath
(a) 2!qo is estimated by the multinomial proportion of deaths in the third in­
terval, so 2lq0 = � = 2 = .40.
(b) S(3) is estimated by the proportion surviving to t = 3, producing
= .35.
so (3) =
(c) q3, a conditional probability, is estimated by the binomial proportion of
deaths in the fourth interval, out of those alive at the beginning of that
interval, so q3 = � = �.
0
t
lo
�EXAMPLE 4.51! Suppose our sample in Example 4.4 is known to be sub­
ject to a survival distribution that is uniform on (0,5]. (a) Calculate the vari­
ance of 2lq0 and the conditional variance of 3 • (b) Estimate these variances
if no distribution is assumed.
p
[SOLUTION!
(a) 2lqo = F(3)-F(2) = .20, under the uniform distribution. Then, by
(.20X.80) = .008. p3 = S(4) = . 0, and n3 = 7,
A ) =
yar ( 2 I %
(4.15,) u
5
20
S(3)
so, from (4.21), Var(p3 1 n3 ) = .03571.
(b) In this case we use 2lq0 in place of 21 qo, and 3 in place ofp3, obtain0
ing Var(2lq0) = .012, and Var(p3 ln3) = .01749.
p
4.4.5 Estimation ofS(t)from {p;}
We recall that S(t) is the unconditional probability of surviving from time O
to time t, and the direct approach to estimating S(t), given by (4.12), is an un­
conditional approach.
As an alternative to estimating S(t) by (4.12), we could use the ap­
proach introduced in Section 2.4.6, producing
A
S(t)
= Po · Pi · · · · · P1-1 ·
A
A
/\
(4.22)
This estimation approach follows logically from the conceptual relationship
S(t)
= Po · Pl · · · · · P1-1 ·
(4.23)
It is easy to see by general reasoning that, although each p; in (4.23) is a con­
ditional probability, the product of them, which is S(t), is unconditional.
Thus S(t) is the same unconditional probability concept, whether estimated
by the direct (unconditional) approach of (4.12) or the indirect (conditional)
90
Tabular Survival Models Estimatedfrom Complete Data Samples
approach of (4.22). Furthermore, for a given sample outcome in the cohort,
complete data study design of this chapter, it is easy to see that the same
numerical value of S(t) will result from both (4.12) and ( 4.22), provided each
in (4.22) is determined by (4.19). The demonstration of this is left as an
exercise.
In studies which are not restricted to an initial cohort, or which al­
low for withdrawals or termination of observation before all have died, the
(4.12) approach to S(t) will not be possible. In these studies, described in
Chapters 5-7, the (4.22) approach to S(t) will be taken.
P;
4.4.6 Estimation of the Hazard Function
Fm our cohort, complete, grouped deaths sit11i1tion, we have estimated each
conditional p, and the survival distribution function S(t). In this section we
will consider the estimation of the HRF of T. The approach here will be to
express the hazard function in terms of a function whose estimation has al­
ready been discussed, and then to substitute the estimator of that function.
Let t" = t+ f represent the midpoint of the interval (t, t+ l]. (Recall
that the HRF relates to a point of time, not an interval.) To express the haz­
ard rate .X(t*) in terms of p1 (or q,), we need to make a distribution as­
sumption, such as one of those described in Section 3.5 in the context of the
life table.
For example, assuming that T is exponentially distributed over
(t, t+ l], so that .\(t) is a constant, we recall from (3.67) that
(4.24)
so ,\(t*) is estimated by
/\
*
/\
>..(t ) = - lnpr
(4.25)
Recall that p1 is estimated by the binomial proportion estimator H,+t
, condi�
tional on the value of n, being given. (Note that N1+1 is the random variable
here.) Thus the estimator for ,\(t*) can be written as
.
(4.26)
Grouped Times ofDeath
91
/\
which is a biased estimator. The random variable>.(!*) is a natural log func"
tion of the binomial random variable N,+1, so the variance of .A(t*}, conditional on n,, can be approximated by the method of statistical differentials
using formula (2.76). This results in
(4.27)
�XAMPLE 4.61! Again using the sample data given in Example 4.4, esti/\
mate>.(3.5). Also estimate the variance of>.(3.5), using the sample outcomes.
(SOLUTION! We will need the sample results p3 = �, q3 = �, n3 = 7.
From (4.25), ..\(3.5) = -lnp3 = 1.94591, and from (4.27), evaluated by sam� /\
6/7
D
)(?) = .85714.
pie results, we have Var[.A(3.5)jn3] = (l/7
/\
/\
4.5 SUMMARY
In this chapter we have introduced the topic of estimating a tabular survival
distribution from sample data. The estimation exercise was fairly straight­
forward here due to the convenient properties of complete data samples.
With the general idea of survival model estimation now familiar to us, we
can deal with the complications created by incomplete data, which we will
do in the remaining chapters.
But first we need to make some final observations on the material of
this chapter.
The observed survival distribution defined and analyzed in Section
4.3 is somewhat special. We can describe it as an exact and complete repre­
sentation, or description, of the survival pattern actually experienced by the
sample. It can be used as an estimate of the operative (but unknown) S(t) to
which this sample was subject. We must become totally comfortable with the
idea that a sample is subject to a survival distribution, which we call S(t),
which is likely to be different from the S0 (t) that the sample actually exper­
iences. We never know how different S(t) and S0 (t) are, since S(t) is forever
unknown. Thus, to summarize, S0 (t) is the precise survival pattern of this
sample, and it plays the role of an estimate of S(t).
In later cases, we will again derive an estimate of S(t), to be called
A
/\
S(t), from sample data, but it will not be correct to call such S(t) the exact
lJ2
Tabular Survival Models Estimated from Complete Data Samples
survival pattern of the sample. That property belongs exclusively to the co­
hort, complete data, exact time of death situation. It is because of this prop­
erty that we use the special symbol S0 (t) in this case; in other cases th e
general S(t) is used.
Finally, it should also be noted that S0 (t), defined by (4.1), does pro­
vide an estimate of S(t) for all t. In light of this, this S0 (t) is in the nature of a
parametric, rather than a tabular, survival model. We introduced it in this
chapter, rather than in Chapter 8, because its simplicity makes it an excellent
first lesson in survival model estimation.
The grouped deaths study design of Section 4.4 is associated directly
with the estimation of a tabular model, that is, one which gives values of S(t)
only for certain, evenly spaced, values oft.
Some observations should be made with regard to the properties of
the estimators developed in Section 4.4.
It should be clear that those estimators which are binomial or multi­
nomial proportions, which include 0 ,
and the unconditional (4.12)
,jq q" p,,
approach to S(t), are· also maximum likelihood estimators (MLE), with the
optimality properties of unbiasedness, consistency, and efficiency, as des­
cribed in Appendix A.
I\
The conditional (4.22) approach to S(t) is through the product of the
MLE's 0 through 1_ 1• S(t) is also an MLE, and is unbiased.
The estimator �(t*}, defined by (4.25), is derived from the MLE Pt
I\
under the exponential assumption. Under this assumption, A(t*) is also an
MLE, but it is biased.
As a final observation on the results derived in this chapter, we note
that to estimate .X(i*) we made the exponential distribution assumption. This
was our first use of a distribution assumption for the purpose of aiding our
estimation work. In the following chapters frequent use of this and other
assumptions will be made.
The results of Section 4.4, and some properties of those results, are
summarized in Table 4.1.
p
p
Summary
93
"Function
tobe
J;stimated Estimator
,fqo
D,
n
TABLE4.1
Unconditional
or Conditional Biasedness
Variance
Unconditional
,lqo(l-,lqo)
Unbiased
n
(Exact)
S(t)
N,
n
Unconditional
Unbiased
S(t) · F(t)
n
(Exact)
P,
N,+1
n,
!2i
n,·
q,
. .\(t
*)
- In N,+1
n1.
Conditional
on n,
Unbiased
Conditional
onn,
Unbiased
Conditional
onn,
Biased
/!J..:..!JJ.
n,
(Exact)
/!J..:..!JJ.
n,
(Exact)
q,
Pt· n1
(Approximate)
4.1 Introduction
4.2 Study Design
4.3 Exact Time of Death
4· l. Let �o represent the observed average lifetime of a cohort whose times
at death are exactly recorded. Find &0 for the sample described in
Example 4.2.
4-2. Let &, be the observed average remaining lifetime for those in the
cohort who survive to time t. Find &6 for the sample of Example 4.2.
94
Tabular Survival Models Estimatedfrom Complete Data Samples
4-3.
Suppose the cohort of Example 4.2 is actually subject to a uniform
survival distribution on (0, 15]. Compare the observed average life.
times �o and �6 to the expected, according to the assumed distribution.
4-4.
We recognize that �o is a random variable, since it depends on the
random lifetimes of then units in the cohort. Find expressions for the
expected value and variance of �0, assuming that the lifetimes of dif­
ferent units in the cohort are independently and identically distrib­
uted.
4-5.
The following values are taken from a standard life table:
f.10 = 80,000
[g4 = 37,000
fg3 = 42,000
Four individuals, each exact age 70, are observed until death. The
deaths of these individuals occur at exact ages 83.30, 83.34, 83.36,
and 83.47. Assuming that the operative survival model for these indi­
viduals is given by the standard table, determine the probability that
the number of individuals whose lifetimes exceed their expected me­
dian future lifetimes is greater than the number actually observed.
4.4 Grouped Times of Death
4-6.
Find Cov(,I q0 , rl fJo).
4-7.
Find Cov [S(t),S(r)], where t < r. Hint: consider the trinomial distribution of the random variables (1) deaths before t, (2) deaths between
t and r, and (3) deaths beyond r (i.e., survivors at r).
4-8.
A cohort of n persons is observed until all die, with deaths grouped in
A
A
fixed intervals. If Var [S(t)] = .0009, Var [S(r)] = .0016, and their coA
A
A
variance is Cov[S(t), S(r)] = .0008, find E[S(t)].
4-9.
Let n = 500. If Var [S(t)] = .000420, Var [S(r)] = .000255, and if
/\
A
S(t) > 2 · S(r), where t < r, then find Cov[S(t),S(r)].
A
A
A
A
95
Exercises
4-l0. Demonstrate that, if each value of /J; is determined by ( 4.19), then the
estimated value S(t) will be the same whether obtained by the approach of (4.12) or by the approach of (4.22).
4-11. A complete mortality study is made on a sample of 7 lives. The co­
hort is believed to be subject to the survival pattern given by the
following life table.
I
0
l
2
3
l,
100
?
10
0
The value of f 1 in the table is not given, but it is known that the un­
conditional variance of D 1 , the random variable for the number of
deaths in (1, 2] for this sample is 1.68. It is also known that, in the life
table, do > d1. Find the probability that the outcome for this sample
will be one death in (0, 1], one death in (1, 2], and five deaths in (2, 3}.
4-12. A complete mortality study was made of n = 1000 persons all exact
age 100. The following is a partially completed record of results, by
duration since age 100.
Duration
(I)
0
l
2
3
4
5
6
7
8
Total
Observed
Deaths
Observed
Survivors
Observed
Conditional
ProbabiHty
Estimate of
Survival
Function
(d,)
(n,)
(q,)
[S(t)J
1.000
150
1000
.100
50
100
.200
.250
200
100
1000
Complete the missing entries in the table.
96
Tabular Survival Models Estimated from Complete Data Samples
4-13. Derive formula (4.27).
4-14. Assume that Tis unifonnly distributed over (t, t+ 1].
(a) Express >.(i*) in terms ofPt ,
(b) Express the estimator >.(t*) in terms of the binomial random variable N1+i· (Note that this is conditional on n1 being given.)
/\
(c) Find the approximate variance of >.(t*).
(d) Is >.(t*) biased or unbiased?
I\
/\
4-15. Show that the expression for Var[>.(t*)] derived in Exercise 4-14(c)
can also be written as
[>.(i*)]2 .
{ I - _41 [>.(1*)]2}.
n1 q,
4-16. Of l 00 persons surviving to time t, four die before time t+ I. Use this
data to estimate the central rate m1, under both the exponential and
uniform distribution assumptions. Also estimate the variances of these
estimators for m1•
4.5 Summary
4-17. Show that
,lq0 is a maximum likelihood estimator.
4-18. Find a general expression for est. Var [S(t)], when the sample estimate is used to estimate S(t).
CHAPTEJR5
TABULAR SURVIVAL MODELS ESTIMATED
FROM INCOMPLETE DATA SAMPLES
STUDY DESIGN
5.1 INTRODUCTION
The situation described in Chapter 4, in which values of the survival func­
ti. on at various times were estimated by the proportion of the study sample
still surviving, was a special situation usually found only in laboratory or
other closely controlled environments. We recall that the properties of that
situation were that all members of the sample came under observation at
time t = 0, none left the sample other than by death, and observation was
continued until all had died. In such a situation we say that the data are
complete.
Many estimation situations encountered in practice, including those
encountered by the actuary in a life insurance or pension setting, do not
exhibit the properties of complete data. Rather it will be common to find per.,
sons coming under observation at times (or ages) greater than zero, and not
all persons will remain under observation until death. This termination of ob­
servation (other than by death) can occur in the one of two ways described in
the next two paragraphs.
In some situations there will be one or more random events (in
addition to death) to which members of the sample are susceptible, the
occurrence of which eliminates the member from the sample. For example,
in a pension plan, active members might be eliminated from the study sam­
ple through disability, retirement, or voluntary termination. In a sample of
life insurance policyholders, members might lapse their policies. In general,
we will collectively refer to all random decrements (other than death) as
withdrawals from the study. It is important to note that withdrawal is a
random event in the same sense that death is a random event.
The longitudinal study design of Chapter 4 is feasible only when the
sample consists of relatively few members with a relatively short lifetime,
such as mechanical devices, laboratory animals, or unhealthy humans. Most
studies conducted by actuaries deal with large samples of healthy humans; to
keep a longitudinal study open until all persons have died is simply not
feasible. Instead a cross-sectional study design is used, in which a period of
98
Incomplete Data Samples: Study Design
observation is chosen, and sample members are observed only during that
period. When the observation period closes, most members are still alive and
in the sample, and observation terminates for them at that time. These mem­
bers are called enders, since their observation terminates as a result of the
ending of the observation period.
It is important to note that whereas withdrawals are random events,
enders are not. By setting a date at which the observation period will end, the
maximum age at which a particular sample member will cease observation is
thereby established. Thus we say that each sample member has a scheduled
ending age. If the member remains in the group to that age, the event of
"ending" is then certain to occur. An example will make this clear.
!!EXAMPLE 5.tlj The period of observation for a sample of pension plan
participants is January I, 1995 to December 31, 1998. Participant A, who
joined the plan in 1990, was born on April I, 1960. At what age does Partici­
pant A come under observation, and at what age is Participant A scheduled
to cease observation?
!SOLUTION! Since Participant A is already in the plan on January 1, 1995,
then observation begins on that date. This person will be age 35 on April 1,
1995, so he comes under observation at age 34.75 on January l, 1995. If
neither death nor withdrawal intervenes, Participant A will be age
· 38.75 on
0
December 31, 1998, so this is his scheduled ending age.
Example· 5. I illustrates the basic information that will be required on
each sample member for the estimation approaches that will be presented in
the next several chapters. This information consists of the following:
( l) Dates of the observation period
(2) Date ofjoining the group under observation
(3) Date of birth
(4) Date of death or withdrawal, where applicable
From items (1), (2) and (3), each sample member's age at the start of obser­
vation and scheduled age at the close of observation can be established. For
the ;th sample member, we will refer to these two ages as y, and z;, re­
spectively. Note that, necessarily, y; < z;. Thus the scheduled age span of
observation is represented by the ordered pair (V;,z,) for the i1h member. If
death or withdrawal occurs while under observation (i.e., before age z;), then
observation will actually end prior to the scheduled ending age. In no case
can observation continue beyond the scheduled ending age.
Introduction
99
[j)XAMPLE 5.21[ The members of a fraternal organization form a study
sample with observation period of July 1, 1990 to June 30, 1996. Member B
was born on June 30, 1943 and joined the organization on January l, 1992.
What is the ordered pair (y;,z;) for Member B?
�OLUTION! Since Member B joined the organization during the observa­
tion period, her observation began at that time, rather than at the start of the
period. On January 1, 1992, Member B was Y; = 48.5. Her scheduled ending
age on June 30, 1996 is z, = 53. Thus her ordered pair is (48.5, 53).
0
A practical question arises as to the computation of fractional ages
whenever the dates have not been so conveniently chosen. Since all large
studies of the type contemplated here will be totally computerized, it will be
an easy matter to count ages in days, and then convert them to fractions of a
year. This is illustrated in the final example of this section and more exten­
sively in Chapter 9.
#EXAMPLE 5.3ij An alumni association membership forms a study sam­
ple. The observation period begins for each member on the member's birth­
day in 1994 and ends on December 31, 1998. Member C is born on July 12,
1968, joins the alumni association in 1991, and dies on March 20, 1998.
What is Member C's ordered pair (y;,z;)? At what age does observation
actually cease for Member C?
!SOLUTION! Observation begins on Member C's birthday in 1994, which
is exactly age y; = 26. Observation is scheduled to end December 31, 1998.
Member C will be age 30 on July 12, 1998. Counting days from there to
December 31 produces 19+31+30+31+30+31 = 172 days, so the scheduled ending age is z; = 30��� = 30.47. However, we see that death inter­
venes on March 20, 1998, on which date Member C has attained age 29
years plus 19+ 31 + 30+31 + 30+31 + 31+29+20 = 252 days, or 29.69 years.
D
This is the age at which observation actually ceases.
5.2 INTERVAL ESTIMATES
The overall approach to estimating S(x), or, equivalently, fx , which we will
use in the presence of incomplete data will be to first estimate Px , the condi­
tional probability of survival to age x+ I, given alive at age x. Then, since
S(x) = po · Pi · · · · · Px- 1, it follows that a natural estimator of S(x) is
A
S(x)
= Po· Pi · · · · · Px-1·
A
A
A
(5.1)
Incomplete Data Samples: Study Design
100
ln turn, we will frequently find it convenient to estimate qx , rather than Px ,
from the sample data, and then to obtain an estimate ofPx as
(5.2)
The generalized estimation interval (x,x+ l] is illustrated in Figure 5.1.
x-2
x
x-1
x+2
!FIGURE 5.11
The notation (x, x+ l] is used to indicate that deaths at age x+ l are assigned
to this interval, whereas deaths at age x belong to the previous interval
(x-l,x]. This assignment is somewhat arbitrary, and indeed the alternative
[x, x+ l) has been used by some writers. We prefer (x, x+ I], since existence
at age x is assumed in the concept of qx as "the probability of death within a
year for a person known to be alive at age x." Thus death at age x is not
possible in this interval.
In order to estimate qx , we need to know what contribution, if any,
each member of the sample makes within the interval (x,x+ I]. In other
words, we need to know the survival experience of each member between
ages x and x+ 1. To obtain this, we need to see how the ordered pair (y;,z;)
for person i relates to the interval (x, x+ I].
First we note that if Zi :; x, then person i makes no contribution to
]
(x, x+ 1 , since he left the study without moving· into that estimation interval.
Similarly, if y; 2: x+ 1, then person i again makes no contribution to the
internal (x, x+ 1], this time as a result of coming under observation after age
x+ l. These cases are illustrated in Figure 5.2.
------ l
Z;
1 ------
OR
x
x+l
y;
!FIGURE 5.2!
If Yi ::; x and z; :::: x+ 1, as shown in Figure 5.3, then person i is scheduled to
be under observation for the entire interval (x, x+ 1 ].
l ----------------------- 1
x
x+l
!FIGURE 5.3]
Z;
Interval Estimates
101
finally, it is possible that x < Yi < x+ I or x < zi < x+ l, or both, as shown
in Figure 5.4.
l -------- l
x
x+l
!FIGURE 5.4f
Disregarding the cases where z; � x or Yi � x+ l, so that person i
does not contribute to (x, x+ I] at all, we can say that person i enters (x, x+ 1]
at age x+r; and is scheduled to exit (x,x+ l] at age x+s;. If y; � x, as in
Figure 5.3, then person i enters (x,x+l] at exact age x, so that r; = 0. If
x < y; < x+ I, as in Figure 5.4, then entry into (x,x+ l] is at age x+r; with
O < r; < I. If z; 2: x+ I, as in Figure 5.3, then person i is scheduled to exit
(x,x+ l] at exact age x+ I , so thats;= 1. If x < z; < x+ 1, as in Figure 5.4,
then scheduled exit from (x,x+ I J is at age x+s; with O < Si < I. Note,
overall, that O � r; < 1 and O < s; � l. That is, r; can be O but not 1, where­
ass; can be I but not 0.
This correspondence between the entry and scheduled exit ages for
the study, represented by (y;,zi), and the ,entry and scheduled exit ages for
the estimation interval (x, x+ I], represented by (x+r;, x+s;), is very impor­
tant. The basic data for the study implies the pair (y;,zi), as illustrated in Ex­
amples 5.1, 5.2 and 5.3. But in order to estimate qx for the interval(x,x+I],
the necessary information is given by (x+r;,x+s;), or, more simply, (ri,s;),
with x being understood. Practice with the conversion from (y;,z;) to (r;, s;),
for a given (x,x+ I], is provided in Examples 5.4 and 5.5, and in the exercises.
It should also be noted that person i might not contribute to (x, x+ l]
even if z; > x. Recall that z; is the age at which observation is scheduled to
end due to the closing of the observation period. If person i actually dies or
withdraws at an age less than x, then no contribution is made to (x, x+ l]
even though such a contribution had been scheduled. For this person, r; and
s1 are not defined for the interval (x,x+ 1].
!!EXAMPLE 5.4lj For Participant A in Example 5.1, what is the ordered pair
(r;,s;) for the estimation interval (38,.39]?
!SOLUTIONI Since y; = 34.75 < 38, then r; = 0. Since z; = 38.75, then
0
S; = .75.
102
Incomplete Data Samples: Study Design
!!EXAMPLE s.sJj For Member C in Example 5.3, what is the ordered pair
(r;,sJ for the estimation interval (30,31]?
!SOLUTION! Here y; = 26, implying r; = 0, and z; = 30.47, implying
s;= .47. However, death occurred at age 29.69, so Member C does not
contribute to (30, 31] at all. Thus (r;,s;) is simply not defined for the interval
(30,31].
D
The approaches to estimating qx which are described in Chapters 6
and 7 are designed to accept ordered pairs (r;,s;) for the estimation interval
(x, x+1] satisfying the general conditions O :S r; < 1 and O < s; :S 1. There
are several special cases, however, under which the estimation mathematics
is simplified.
If, for a certain (x,x+I], r;= 0 ands;= 1 for all i, so that all per­
sons enter the estimation interval at its beginning and are scheduled to re­
main under observation to age x+1, the estimation of qx is considerably sim­
plified. We call this convenient situation Special Case A.
If, for a certain (x, x+1],s;= 1 for all i but the general O :S r; < 1
exists, then persons can enter (x, x+1] at any age but all are scheduled to exit
the interval at age x+1. This is referred to as Special Case B.
If, for a certain (x, x+1], r; = 0 for all i but the general O < s; :S I
holds, then all persons enter (x, x+1J at age x but have scheduled exit ages
scattered throughout the interval. This is called Special Case C.
The significance of these special cases,in terms of the simplification
that they introduce to the process of estimating qx , will be illustrated in Chap­
ters 6 and 7. In particular, the simplification provided under Special Case A
is so delightful that we are sometimes motivated to design studies in a man­
ner so as to produce this special case. This issue, in an actuarial context, is
explored in Chapter 9.
How are Special Cases A, B and C obtained? One way is by group­
ing, or averaging, values of y; and z;. For example, for all y; such that
x - } < y; < x+}, we could substitute x for the actual value of each y;.
Then all persons would be entering observation at an estimation interval
boundary, so r; would be zero for all i. Similarly, all values of z; such that
x - } < z; < x+}, could be replaced by x. Then all scheduled exits would
be at interval boundaries, and we would have s;= l for all i. This grouping
approach to estimation simplification will be pursued in Chapter 9.
Even without grouping, the various special cases can be obtained as
a result of the definition of the observation period. In Example 5.3 we saw
that having an observation period begin on a member's birthday necessarily
resulted in an integral value for y;. Then if x is also an integer, it follows
Interval Estimates
103
that all Yi occur at estimation interval boundaries, so that r; = 0 for all i
Similarly, s; = l for all i if the observation period ends on the member's
birthday. If the observation period opens on a fixed date for all members,
then r; > 0 for some i; if it closes on a fixed date for all members, then
s; < 1 for some i.
[iXAMPLE 5.6!1 An observation period opens on January l , 1995 and will
close, for a given member, on that member's birthday in 1996. Three mem­
bers, all in the study group as of January l , 1995, have the following dates of
birth :
Member
I
2
3
Date of Birth
April l, 1974
July l, 1974
October l, 1974
Which special cases are created for estimation intervals (20,21] and (21,22]?
!SOLUTION! The ordered pairs for the study are (y 1 ,z 1) = (20.75,22),
{yi,z2) = (20.50,22), and (y3,z3) = (19.25,21). For the estimation interval
(20,2 1], the values of (ri ,si) are (.75,1),(..50,1) and (0, 1), for i = 1,2,3,
respectively, which is Special Case B since s; = 1 for all i but r; > 0 for
i = I and i = 2. For estimation interval (21,22], Member 3 is not involved,
and we have (r1,s1) = (0,1) and (r2,s2) = (0,1). Since all r; = 0 and all
s; = 1, we have Special Case A for this interval. The data of this example
are shown in Figure 5.5.
2 ! -------------,
! ---------------l ---------------------:!.
l
21
22
19
20
3
!FIGURE 5.51
D
5.3 SINGLE-AND DOUBLE-DECREMENT ENVIRONMENTS
We have established that the i'h person in a study sample enters the estima­
tion interval (x, x+ l] at age x+r;, 0 :::; r; < l , and is scheduled to exit the
interval at age x+s;, 0 < s;:::; l, as shown in Figure 5.6.
104
Incomplete Data Samples: Study Design
l----------x
x+l
!FIGURE 5.6!
Suppressing the subscript i for convenience, we note that the proba­
bility of survival to age x+s for this person is s-rPx+r, and the probability of
death while under observation in this interval is
s-rqx+r = I - s-rPx+r·
(5.3)
Note that these probabilities are conditional on being alive at age x+r. The
density function for death at age x+t, given alive at age x+r, is given by
(5.4)
This notation is adequate, however, only if the study is being con­
ducted in an environment in which death is the only random event to which
sample members are subject, i.e., if we are working in a single-decrement
environment. If both death and withdrawal are random events, so that we
are in a double-decrement environment, the estimation mathematics is more
complex. In this section we will review basic double-decrement theory to the
extent needed for our estimation work. (For a more thorough review of this
theory, the reader is referred to Chapter 14 of Jordan [41] or Chapter 11 of
Bowers, et al. [12).)
We begin with the basic function ,Px , which has been understood to
represent the probability of not dying before age x+t, given alive at age x, in
a single-decrement environment. When a life age x is in a double-decrement
environment, however, it is necessary to specify the random event for which
,Px represents the probability on nonoccurrence before age x+t. We do this
by using iP� (d) for the probability that death will not occur prior to age x+t,
and by using 1p�(w) for the probability that withdrawal will not occur prior to
age x+t, where each probability has its meaning in its own single-decrement
environment.
Note that we use ,p:(d) to represent exactly the same concept as
represented earlier by the unadorned ,Px· We have simply added the extra
notation to identify the idea of "survival against death." Similarly, we will
now use q�(d) to mean the probability of death in (x,x+ l], and note that q�(d)
represents exactly the same concept as represented by the simpler qx used
heretofore in the text.
Single- and Double-Decrement Environments
105
Associated with ,p:(d) and q:(d) is the force of mortality µ�t (previ­
ously denoted as µx+t when it was clear that it represented the force of
mortality). The reader should recall the relationships
,p:<d)
1'
= exp [ -
and
µ�tdu]
J.'
q', CdJ = I - exp [ -
µ;1.
(5.5a)
ch,].
(5.6a)
Now that withdrawal is also viewed as a random event, there exists a
force of withdrawal, µ�:>,, at age x+t, and the relationships
,p:(w) = exp [ -
and
11
µ�1u du ]
J.'
q;cw> = I - exp [ -
µ;";. du
(5.Sb)
l·
(5.6b)
If the random events death and withdrawal are independent, then
I (d)
(r)
, (w)
rPx = tPx · · rPx
(5.7)
r) (d)
rPx( µ x+t
(5.8a)
r) (w)
tPx( µx+t·
(5.8b)
is the probability of neither dying nor withdrawing prior to age x+t, given
alive and not withdrawn at age x. In this text we will assume the indepen­
dence of these events. The reader should be aware that this independence is
not always assumed in basic multiple-decrement theory (see, for example,
Benjamin and Pollard [11], or Bowers, et al. [12]).
The density functions for death and withdrawal, respectively, at age
x+t, given alive at age x, are
and
The more general relationships
and
s-rq:�} = l - s-rP:�1 = 1 - exp [ I (w)
s-rqx+r
s
l µ�1 du]
[ l
u
]
(w)
µx+udu
= 1 - s-rPI (w)
+r = 1 - exp rs
:x
(5.9a)
(5.9b)
106
lllcomplete Data Samples: Study Design
will arise in our estimation work, as will the relationship
r
()
t-rPx+r
=
d)
t w)
( · t-rPx+r
(
l-rPx+r
I
(5.10)
and the density functions
(5.1 la)
for death and
(T)
(w)
(5.1 lb)
t-rPx+r µx+t
for withdrawal.
w
Finally, we let q�d) and qi ) represent the probabilities of death and
withdrawal, respectively, prior to age x+ l for a life age x, in the presence of
the other decrement (i.e., in the double-decrement environment). Note how
these two probabilities differ in concept from q:(d) and q:(w)_ Mathematic­
ally,
and
q(xw)
1
1 p(r) (d) du - Jo[ (1 q'(d)] [1 q'(w)] (d) d
(5.12a)
1
1
) u= 1 [l- q'(d)] [1- q'(w)]µ(w) du
= 1 up(r)µ(w
u x
u x
x+u
·
x
x+u d
(5.12b)
qx(d) =
O
1
u x µx+u
-u x
-
0
-u x
µx+u U
0
The probability of surviving from age x to age x+ 1 is
(5.13)
More generally, s-rq�t and s-rl1r represent the probabilities of death and
withdrawal, respectively, prior to age x+s for a life age x+r, in the presence
of the other decrement. Corresponding to (5.12a), (5.l2b), and (5.13) we
have
(5.14a)
s
s-rl1r - i u-rP�1rµ�»;udu
s
- i
[1-u-rq:i1] [ 1-u-rq:t)] µ�v:Ju du,
(5.14b)
Single- and Double-Decrement Environments
and
107
(5.15)
Of special interest to us are the relationships between the double­
w
decrement probabilities qid), qi ) and the corresponding single-decrement prob­
abilities q:(d)' q:(w)under the uniform and exponential distribution assump­
tions defined in Chapter 3. These relationships will then be utilized in the
estimation problems encountered in Sections 6.3.2, 6.3.3, 7.4.5, and 7.4.6.
5.3.1 Uniform Distribution Assumptions
Let each decrement, death and withdrawal, have uniform distributions over
(x, x+ 1 ], so that
(5.16a)
and
(5.16b)
as already established by Equation (3.54). By Equation (3.57), we now have
(5.17a)
and
q'(w)
µ(w) x+s - 1-s·q:<w)·
(5.l 7b.)
To find q�d) in terms of q:<d) and q:<w} under the uniform assumptions, we
substitute (5. l 6a), (5. l 6b), and (5.17a) into (5. l 2a), obtaining
1
= q :<dlfo (I-u·q:<w))du
= q:<d)[t-!·q:<w>]-
(5. l 8a)
108
Incomplete Data Samples: Study Desig11
Similarly, using (5. I 2b) we find
(5.18b)
To find q:(d) in terms of q�d) and q� >, we solve the pair of equations
given by (5.18a)and (5.18b). From (5.18a) we have
w
(5.I9a)
and from (5.18b) we have
(5.19b)
Substituting (5.I9b) into (5.I 9a) leads to the quadratic equation
which solves for
(5.20a)
where b = I-! qi ) + ! qid). (Note that the positive radical is not taken since
it would lead to q:(d} > I.) By considerations of symmetry, we have
w
qx1(w) = c � '\/fc2 _ .2. .. qx(w) '
(5.20b)
5.3.2 Exponential Distributioll Assumptions
If both death and withdrawal are assumed to have exponential distributions
(constant forces) over (x,x+1 J, then
and
!(
e-s·/dl
sPx d) =
(5.2Ia)
= e-s·p(-.) '
(5.2Iti)
t(w)
sPx
Single- and Double-Decrement Environments
109
as already established by Equation (3.64a). By the constant force property,
we now have
(5.22a)
and
(w) = (w)
µ
µx+s
(5.22b)
To find qid'J in terms of µ<d'J and µ<w) under the exponential assumptions, we
substitute (5.2la), (5.2lb), and (5.22a) into (5.12a), obtaining
)
- µ(d)11 e -u(µ(d)+µM du
µ(d)
<d) 1 ]
[
(d) + (w) 1-e-(µ +µ'" >
µ
µ
(5.23a)
Similarly, using (5.12b) we find
(w)_ _
µ(w)
}- - <d) µM
qx - µ(d) + µ(w) [ e (µ + )
]
(5.23b)
To find q�(d)in terms of qid) and qi >, we solve the pair of equations
given by (5.23a) and (5.23b). From (5.23a) we have
w
<d) M
} - e-(µ +µ )
µ<d)+µ<w)
(5.24a)
and from (5.23b) we have
(5.24b)
Together (5.24a) and (5.24b) imply
so that
110
Incomplete Data Samp les: Study Design
(w)
µ (d) _i_ qx
( d)
I
qx
(d)
(w))
. µ<d) = µ(d) ( qx +qx
(d)
qx
•
(5.25)
Then by substituting (5.25) into (5.23a) in two places we have
or
l - qx(d) - q(xw) '
which solves for
(d)
(cl)/
( )q,
- In ( l -qx -qxw)
(d)
(w)
(q, +q, )
•
(5.26)
Finally, we have
which can also be written as
•
(5.28a)
•
(5.28b)
By considerations of symmetry, we also have
1(w) _
qx
-
1 _ ( (r))qr
Px
(w>;c (cl}
qr +q,(•))
5.4 SUMMARY
The purpose of this chapter has been to provide an orientation to the problem
of estimating qx over the interval (x,x+ 1], given alive at age x, from which
an estimate of S(x) can then be produced.
Our approach to the estimation of qx is to first process the basic
information on the i1h life in the sample to obtain the ordered pair (y;, z;), the
entry age and scheduled exit age for person i for the study as a whole. (It is
Summary
111
important that the concept of scheduled exit age be clearly understood.) The
pair (yi,zi) then implies the pair (x+ri,x+si), the entry age and scheduled
exit age for person i with respect to the estimation interval (x, x+ 1 J. The
importance of the distinction between entry and scheduled exit ages for the
study as a whole, and those for a particular estimation interval, cannot be
overemphasized.
We saw how certain definitions of the observation period led to con­
venient values of ri and si, which, in tum, led to simplified estimation situ­
ations. We also saw, briefly, how this simplification could be achieved by
grouping, or averaging, neighboring values of Yi and z;. This practical ap­
proach to large-sample studies will be explored further in Chapter 9.
Finally, we reviewed basic double-decrement theory for the purpose
of becoming familiar with the necessary relationships and notation before we
encounter estimation in a double-decrement environment in later chapters.
We will then be able to concentrate on the estimation theory without being
diverted by the development of the basic mathematics.
S.5 EXERCISES
S.1 Introduction
5-1 .
The members of a pension plan form a study sample over the observa­
tion period January 1, 1992 to December 31, 1997. Member D was
born April 1, 1966. What other information is needed in order-to
obtain the ordered pair (yi , zi ) for Member D?
5-2.
Find the value of Yi for Member D of Exercise 5-1, if he joined the
pension plan on (a) April 1, 1991; (b) April I, 1994; (c) April 1,
1998.
5-3.
Find the value of zi for Member D of Exercise 5-1 if he joined the
pension plan on April 1, 1991, and if he (a) is still alive and in the
plan on December 31, 1997; (b) dies on September 23, 1997.
5.2 Interval Estimates
5-4.
Under what condition on Yi is it true that Yi = x+r;?
5-5.
Under what condition on Zi is it true that z; = x+s;?
112
Incomplete Data Samples: Study Design
5-6.
Assume that Member D of Exercise 5-1 joins the pension plan on
April 1, 1991, and dies on September 23, 1997. Find the ord ered pair
(r;,s;) for each of the following:
(a) Estimation interval (25, 26]
(b) Estimation interval (28, 29]
( c) Estimation interval (31, 32]
5-7.
Which of the special cases defined in the text is represented by each
of Exercise 5-6 (a), (b), (c)?
5.3 Single- and Double-Decrement Environments
5-8.
r
If q:(d) = . 10 and q�(w) = . 20,' find qi )_
5-9.
If the force of mortality is given by µid) = 3(lOt-x)' and the force of
withdrawal is given by µiw) = 3(I06-x), find the conditional density
function for death at age 80+t, given alive at age 80.
5-10. If both µ�t, and µ���, are constant over O < t < 1, find q�1 given
v
that q�0(d) = q�/ ) = .20.
5-11. Using Equations (5.20a) and (5.20b ), verify that
(1- qxt(d)) (l- qx
1(w)
)
- Px
_
(r)
under the uniform distribution assumptions.
5-12. Using Equations (5.28a) and (5.28b), verify that
= Px(r)
under the constant force assumptions.
1
5-13. Show that qid) = qirl ( :����) under the constant force assumptions.
1
'r
( w)
(d) + J.lx+t·
5-14. Show that µ�+1) = µx+t
CHAPTER6
TABULAR SURVIVAL MODELS ESTIMATED
FROM INCOMPLETE DATA SAMPLES
MOMENT PROCEDURES
6.1 INTR ODUCTION
The foundation which underlies estimation by a moment procedure is the
statistical principle that the expected number of deaths to be observed in
(x, x+ I] from a certain sample should be equated to the number actually
observed. Assuming that the number of actual deaths in the age interval, to
be denoted by dx, is readily available, then the estimation problem consists
of two steps:
(1) Finding an expression for the expected number of deaths in
(x,x+I];
(2) Solving the moment equation for the desired estimate.
Generally a simplifying assumption will be required to solve the moment
equation.
6.2 MOMENT ESTIMATION IN A SINGLE-DECREMENT
ENVIRONMENT
Assume that the basic data regarding an observation period, dates of birth
and membership into the group under observation, and dates of death have
been analyzed to produce the ordered pair (x+r;,x+s;) for person i's sched­
uled contribution to (x, x+ l], as illustrated in Figure 5.6. Recall thats;= 1
is possible buts; = 0 is not, whereas, r; = 0 is possible but r; = 1 is not.
For person i, the probability of death while under observation in the
single-decrement environment is the conditional probability of death before
age x+s;, given alive at age x+r;, which is s;-r;qx+ri, and this is also the ex­
pected number of deaths from this sample of size one. This is easily
seen, since the number of deaths is one (with probability s;-r;qx+rJ, or
zero (with probability s;-r;Px+rJ. Thus the expected number is
(6.1)
114
Incomplete Data Samples: Moment Procedures
6.2.1 The Basic Moment Relationship
If nx is the total number of persons who contribute to (x, x+ 1 ], then the total
number of expected deaths is L s;-r;qx+r,· (For convenience we will use n
nr
i=I
for nx in our summations.) When equated to the actual number of observed
deaths, we obtain the moment equation
E[Dx]
= L s;-r;qx+r;
i=I
(6.2)
where Dx is the random variable for deaths in (x, x+ l ], and dx is the ob­
served number.
To solve (6.2) for 011r estimate of qx , we will use the approximation
(6.3)
Then (6.2) becomes
qx ·
L (s;-r;) - dx,
i=I
(6.4)
from which we easily obtain
(6.5)
the general form of the moment estimator in a single-decrement environ­
ment.
6.2.2 Special Cases
If r; = 0 ands;= 1 for all nx persons who contribute to (x,x+l], then we
have s;-r;qx+r; = qx , and (6.2) becomes
(6.6)
so that (6.5) becomes
(6.7)
Recall that this is Special Case A, as defined in Section 5.2.
Moment Estimation in a Single-Decrement Environment
115
We recognize (6.7) as the binomial proportion estimator already
encountered in Chapter 4. We also recognize it as the maximum likelihood
estimator of the conditional mortality probability qx , where the model for the
likelihood is a simple binomial model. Thus the number of persons in the
sample, nx , can be thought of as a number of binomial trials. The standard
characteristics of a binomial model are assumed to apply. Thus each trial is
considered to be independent, and the probability of death on a single trial
(qx) is assumed constant for all trials. In such a situation, the sample pro­
portion of deaths, which is given by (6.7), is a natural estimator for this
parameter qx .
Ifs; = I for all, but r; > 0 for some of the nx persons who contribute
to (x,x+ I], then we have s,-r,qx+r, = 1-r,qx+r,, and (6.2) becomes
E[Dx]
=
1-r,qx+r, = dx.
L
i=l
n
(6.8)
The general approximation given by (6.3) then becomes
(6.9)
which is the same as (3.77). Substituting (6.9) into (6.8), we obtain the
result
,
(6.10)
the moment estimator for Special Case B.
If r; = 0 for all, buts; < I for some of the nx persons who contribute
to the interval (x,x+ I], then we have s,-r,qx+r, = s,qx , and (6.2) becomes
E[Dx]
n
= Ls,qx = dx .
i=I
(6.11)
The general approximation given by (6.3) then becomes
(6.12)
which is the same as (3.54). When (6.12) is substituted into (6.11) we obtain
the result
Incomplete Data Samples: Moment Procedures
116
(6.13)
the moment estimator for Special Case C.
!!EXAMPLE 6.l!j From the sample of five persons whose basic data are
given below, estimate q30, using an observation period of calendar year
1997.
Date of Birth
July 1, 1967
April 1 , 1967
January 1, 1967
July 1, 1966
April l, 1966
Person
1
2
3
4
5
Date of Death
-
October 1, 1997
-
April 22, 1997
!SOLUTION! The data are shown in Figure 6.1:
•
•
•
30.25
•
30.50
•
30.75
30
I --------------------- •
2 ______________________________,
31
3------------®---·
4______•
5 ___•---®
!FIGURE 6.1!
The ordered pairs (ri,s;) are (0, .50),(0,.75),(0, 1),(.50, 1), and (.75,l), for
i = 1,2,3,4,5, respectively. Thus the denominator of Estimator (6.5) is
.50 + .75 +. l.00 + .50 + .25 = 3.00. The numerator of (6.5) is d_'( = 1, be­
0
cause person 5 died after reaching age 31. Thus q30 =
f.
6.2.3 Exposure
The concept of exposure was defined in Section 3.3.5 in the context of a life
table model. At this time, and at several other places throughout the text, we
will make use of this concept in the context of a study sample.
------ ---------------------
Moment Estimation in a Single-Decrement Environment
117
The ordered pair (r;, s;) for person i represents the interval of age,
within (x,x+ l], over which person i is potentially under observation. We say
that person i is exposed to the risk of death from age x+r; to age x+s;.
Numerically, s;-r; is the length of time (in years) that person i is exposed, so
it also represents the amount of exposure, measured in units of life-years,
contributed by that person. The total exposure contributed by the sample is
n
I: (s;- r;), the denominator of Estimator (6.5).
i=d
The exposure interpretation of the denominators of Estimators (6. 7),
(6.10), and (6.13) is the same as that for Estimator (6.5).
It is important to recognize that the denominator of the moment
estimator represents the scheduled exposure of the sample. When a sample
member dies while under observation, not all of the scheduled exposure is
realized. Henceforth in this text we will refer to the realized exposure as
exact exposure, to distinguish it from scheduled exposure. Note that exact
exposure cannot exceed scheduled exposure.
!!EXAMPLE 6.21l Calculate the exact exposure, for the interval (30,31], of
the sample described in Example 6.1.
!SOLUTION! Only person 3 has exact exposure less than scheduled expo­
sure, and it is less by .25 years. Thus the total exact exposure of the sample
D
is2.75.
Exact exposure plays an extremely important role in estimation, as
we shall see in Chapter 7. Intuitively, the observed deaths divided by exact
exposure provides an estimate of the central death rate, mx , for the interval
(x, x+ I]. From the derived value of fnx , we could then estimate qx by
A
qx
=
A
mx
.
1+l
2 fr,x '
(6.14)
which uses the linear distribution assumption for fx+s· Alternatively, if we
assume an exponential fx+s (the constant force of mortality assumption),
then this constant force, µ, is the same as mx , so that µ
t,x , and thus
(6.15)
In the next chapter we will see that thisµ (and thus qJ are maximum likeli­
hood estimators.
Incomplete Data Samples: Moment Procedures
118
To summarize, scheduled exposure is divided into observed deaths
to estimate qx in a single-decrement environment by the moment estimation
approach, coupled with the approximation s-rqx+r � (s-r) · qx. Exact expo­
sure is divided into observed deaths to estimate mx , or to estimate a constant
force of mortality over (x, x+ 1]. This latter case will arise regularly in
Chapter 7.
6.2.4 Grouping
Consider Special Case B in which the nx persons who contribute to (x,x+ l]
enter this interval at various ages, but all are scheduled to exit the interval at
age x+ 1. That is, all are scheduled to survive (x, x+ l]; none are scheduled
to be study enders at an age z; such that x < z; < x+ l. It may reasonably be
assumed that many of these nx persons enter the interval at exact age x (i.e.,
have r; = O); let us say that bx is the subset of nx which has r; = 0. (bx is
chosen to stand for beginner, since these persons are exposed from the
beginning of (x,x+ 1 ].)
The remainder of the sample, say kx = nx - bx , all have r; such that
0 < r; < l. Let us assume that the average value of these r;'s is r,
0 < r < 1. (Alternatively, it might be possible that the study was designed
such that kx persons all had r; = r, and no averaging is needed.) This
situation is illustrated in Figure 6.2.
---........)·
kx 1---•'
x
x+r
x+l
IFIGURE 6.21
As a result of this grouping, the expected number of deaths in the
interval (x,x+ I] is bx · qx + kx · 1-rqx+r, so the moment equation becomes
E[Dx]
= bx · qx+kx · l-rqx+r = d_y ,
(6.16)
Under the hyperbolic assumption, l-rqx+r = (1-r) · qx , so (6.16) is easily
solved for the estimator
(6.17)
The exposure interpretation of the denominator of (6.17) is readily apparent.
Moment Estimation in a Single-Decrement Environment
119
Next we consider Special Case C, in which all nx persons are ex­
posed from age x (i.e., haver;= 0), but are scheduled to exit (x,x+l] at
various ages z; such that x < z; � x+ 1. Let Cx represent the subset of nx
scheduled to exit at z; < x+ 1 (i.e., S; < I); the remainder, nx -Cx , are
scheduled to remain under observation to age x+ I.
Let us assume that the average value of thes;'s for the ex subset iss,
O < s < I. This situation is illustrated in Figure 6.3.
nx - Cx l ---------­
Cx l -----•
x+s
x
x+l
!FIGURE 6.31
The expected number of deaths in (x,x+ l] is now (nx-cx ) · qx + Cx • sqx , so
the moment equation becomes
(6.18)
Under the linear assumption, sqx = s · qx , (6.18) is easily solved for.the esti­
mator
nx - (l-s) · Cx ·
(6.19)
Again the exposure interpretation of the denominator is apparent.
Our Special Case C is one that frequently arises in clinical studies.
As seen in Chapter 4, these studies often are designed so that all members of
the study sample come under observation at time t = 0. It follows that all
members who reach the estimation interval (t, t+ I] must enter it at exact time
t, so that r; = 0 for all i. But if the study is terminated before all have died,
then there will be enders at various times, so thats; < I for some i. Because
of the frequent occurrence of Special Case C in practice, we will refer to it
several more times as we develop the theory of estimation in this text.
6.2.5 Properties ofMoment Estimators
In this section we will explore the bias and the variance of the moment
estimators derived thus far in the chapter. It is assumed that the reader is
familiar with these two properties of estimators. Formal definitions of them
are given in Appendix A.
Incomplete Data Samples: Moment Procedures
120
A characteristic of the moment estimators for qx which we will
derive is that they are always unbiased under the assumption, or other ap­
proximation, used to derive them. We will demonstrate this for the general
form of the moment estimator given by (6.5).
To investigate bias, we need to compare the expected value of the
estimator with the true value of the parameter, qx , which it seeks to estimate.
We are using qx to represent the realized value of the estimator random
variable when the random variable for number of deaths, Dx , has realized
value dx. Thus we use Qx for the estimator random variable, so that
(6.Sa)
/\
is the relationship between the random variables Qx and Dx for the general
single-decrement moment estimator. Then the expected value of Qx is given
I\
� s,-r,qx+r,· When s,-r,qx+r, IS apby E [QJ = nE[Dx] ' where £[Dx]
L.,
Z: (s;-r;)
i=I
;�1
proximated by (6.3), we obtain
n
n
L s,-r,qx+r,
i=I
qx · L (s;-r;)
=
n
I:(s;-r;)
i=l
n
'E(s1 -r;)
i=I
- qx .
(6.20)
i=I
This shows that Estimator (6.S ) is unbiased under the approximation used to
derive it. It is biased to the extent that (6.3) deviates from the mortality dis­
tribution to which the sample is actually subject. This last point is illustrated
in the following example.
IIEXAMPLE 6.3!j Analyze the degree of bias contained in Estimator (6.10)
for Special Case B.
!SOLUTION! For Special Case B, we have E[Dx] = I: 1-r,qx+r, and the es/\
timator Q
x =
n
D'
_E (1-r;)
1=1
•
Then E[Qx] = n
/\
E[Dx]
E (l-r;)
1=1
i=l
= ·=�
n
_E 1-r;qx+r;
E (l-r;)
i=I
. If we assume
that 1-r,qx+r, = (1-r;) · qx , then E[Qx] = qx , so Estimator (6.10) is unbiased
under this assumption. However, as shown in Chapter 3, this assumption
implies that the force of mortality decreases over (x, x+ l ). Thus if the force
is constant or increasing, as is true over most ages other than juvenile ages,
/\
Moment Estimation in a Single-Decrement Environment
121
n
then 1-r;qx+r, > (1-r;) · qx . In this case we have E[Dx] > E (1-rJ · qx , and
i=l
E
(l-r;}q
x
/\
= qx, which shows that, for most ages, Estimator
thus E[Qx] > 1= 1•
n
E (1-r;)
i=l
(6.10) is positively biased.
0
To detennine the variance of the general moment estimator given by
(6.5), note first that the random variable for number of deaths in (x, x+ l],
Dx , can be written as
n
Dx = LD; ,
(6.21)
i=I
where D; is the random variable (either O or l) for number of deaths from
person i. We hypothesize that all D;, for i = 1,2, ... ,n, are mutually inde­
pendent. Since D; is binomially distributed, with sample size 1 and proba­
bility of occurrence s;-r,qx+r,, we have
Var(Dx) =
L Var(D;) - L si-r,q +r (1- s -r;q +r; ), (6.22)
n
n
i=I
i=I
so that
x
1
1
x
n
=
I: s,-r;qx+ri( l - S;-r;qx+rJ
i=I
[ I)s;-r;)] 2
(6.23)
r=I
/\
The notation Var(Qx lnx , {r;,s;}) is used to remind us that the variance is conditional on the size of the sample, nx , and the set of times at which persons
enter and are scheduled to exit (.x, x+ l]. Expression (6.23) could then be
simplified by using approximation (6.3).
Alternatively, the variance of (6.5) can be approximated by noting
n
"
that Qx is approximately a binomial proportion with sample size _E (s;-r;).
r=I
Thus the approximate variance is
/\
Var(Qx lnx , {r;,s;}) �
I: (s;-r;)
II
i=I
(6.24)
122
Incomplete Data Samples: Moment Procedures
�xpressions for the variances of Estimators (6.7), (6.10) and (6.13),
for Special Cases A, B, and C, respectively, are analogous to Expressions
(6.23) and (6.24), and their derivations are pursued in the exercises.
!!EXAMPLE 6.411 Find an exact expression for the variance of the Special
Case C estimator given by ( 6.19).
!SOLUTION! Let D; be the random variable for deaths out of the ex
group, and D;' be the random variable for deaths out of the (nx -cx) group.
Note that D x = D; + D;', that D; and D;' are both binomial random
variables, and that they are independent. From these properties we find that
Var(Dx) = Var(D;) + Var(D;') = Cx · 5 qx(l- 5qx ) + (nx -Cx) · qx(l-qx), so
I\
Cx·,qx(l-,q,)+(nx -cx )·qx(l-q,)
0
that Var(Qx In.x, C)
x =
[nx - (l-s)·cxJ2
6.2. 6 A Dijferent Moment Approach for Special Case C
We saw in Section 6.2.4 that the expected deaths out of the c.x , all scheduled
to exit at age x+s, was given by C.x · sqx. The fractional-interval probability
required an assumption (such as the uniform) in order to solve the resulting
moment equation. As an alternative to making such a distribution assump·
tion, we now consider a different approach.
Assume that the Cx scheduled enders are subject to the same proba­
bility of death in (x,x+ l] as are the remaining nx -Cx , namely qx. Now con­
sider all of the deaths out of C.x in (x,x+ 1]; let us call this d;. Some of these
deaths occur before exit from the study, and are thus observed, whereas
others occur after exit, and are not observed. Let d; denote the number of
observed deaths out of ex , and let a: represent the proportion of d; that are
observed. Thus d; = a· d;.
Note well that d; is not known, so a is not calculated. Rather, a: is
an input parameter, presumably derived from other data, or merely assumed.
The assumption of a: takes the place of a distribution assumption.
Of the Cx group, Cx · qx are expected to die in (x, x+ 1 ], and a: · Cx · qx
is the expected number of observable deaths. (nx -C.x) · q.x is the expected
observable deaths out of nx -Cx , so total expected observable deaths, equated
to the actual observed deaths, gives the moment equation
A person scheduled to exit observation at age x+s as a study ender,
and who survives to that age, is then observed to be an actual ender. Let ex
denote the number of actual enders in the sample. Now let us consider the
Moment Estimation in a Single-Decrement Environment
123
expected number of observable enders in the sample. We have cx (l-qx) not
expected to die in (x,x+ 1) at all, and (1-a) ·ex · qx expected to die, but
after departure so that they are observed enders. Then total expected observ­
able enders, equated to actually observed enders, gives the moment equation
(6.26)
Note that there are no observed enders in (x, x + 1] from the nx-Cx group.
The estimator of qx is derived by eliminating Cx from Equations
(6.25) and (6.26). From Equation (6.26), we have Cx - 1 -exo:· ; substituqx
ting this into (6.25) gives us
nx . qx _
(1-a:)· ex · qx
- dx ,
1 _ 0 . qx
which leads to the quadratic equation
a · nx · q; - [nx - (1-a:) · ex + a· dx]qx + dx - 0,
the solution of which is
b-
Jb - 4a · n · d
2
2a·nx
x
x
(6.27)
(6.28)
where b = nx - (1-a:) ·ex + a· dx. The negative radical is used in (6.28)
because the positive radical would result in qx > I.
It turns out that Estimator (6.28) is also a maximum likelihood
estimator, which we will show in Chapter 7 (see Exercise 7-9). We also
defer to Chapter 7 the derivation of the variance of this estimator (see Exer­
cise7-24).
One advantage of Estimator (6.28) over Estimator (6.19) is that
(6.28) does not require knowing the value of ex, but rather depends on the
number of actually observed enders, ex. Also, it has some of the desirable
properties of an MLE. Although it does not require a distribution assump­
tion, as does ( 6.19), it does require an assumed value for a.
This estimator was derived by Schwartz and Lazar [66).
IEXAMPLE. 6.S!! In a Special Case C study, nx = 1000, ex = 100 (all of
whom are scheduled to be enders at age x+!), and dx = 20, of which d; = 2.
What value of a will produce the same estimate of qx by Estimator (6.28) as
that produced by Estimator (6.19)?
124
Incomplete Data Samples: Moment Procedures
!SOLUTION! Estimator (6.19) gives qx= 1000_z_�S)(IOO) = 1�5. Since
ex= Cx - d; = 98, then b = 1000-98+98o+20o: = 902+ 1180:. It is easier
to solve for o: from Equation (6.27) than from Equation (6.28). Substituting,
2
we have (1000a)(1:5) -(902+118a)(ii5)+20=0, which produces
o:= .23864. Since a is the proportion of deaths out of Cx which occur in the
first quarter of (x,x+ l ], this result, which is close to .25, is very reasonable.
0
6.3 MOMENT ESTIMATION IN A DOUBLE-DECREMENT
ENVIRONMENT
As in the case of estimation in a single-decrement environment, we assume
that the sample data have been analyzed to produce the ordered pair
(x+r;, x+s;) for person i's scheduled contribution to (x,x+ 11, as illustrated
in Figure 5.6.
The probability of death for person i while under observation in the
double-decrement environment is s;-r,q�1r,, and this is also the expected
number of deaths from this sample of size one, as demonstrated by (6.1) for
the single-decrement case.
6.3.1 The Basic Moment Relationships
As explained in Section 5.3, we now consider both death and withdrawal to
be random events. Thus, analogous to the moment Equation (6.2) in the
single-decrement case, we now write the pair of moment equations
E[Dx]
and
= L s;-r,q�t; = dx
n
(6.29a)
i=l
(w)
= Wx
E[Wx] = I: S;-riqx+r,
n
i=l
(6.29b)
where Wx is the random variable for withdrawals in (x, x+ l ], and Wx denotes
the observed number of withdrawals in the sample.
A simple way to solve Equations (6.29a) and (6.29b) for estimates of
w
the double-decrement probabilities qid) and qi ) would be to use approxima­
tions analogous to (6.3), namely
Moment Estimation in a Double-Decrement Environment
125
(6.30a)
and
(w)
S;-r,qx+r;
�
� (s--r-)
X ),
1
l · iw
(6.30b}
dx
I: (s;-ri)
(6.3 la)
Wx
E(s;-r;)
(6.31b)
producing
A(d)
qx -
n
i=I
and
A(W)
qx
=
n
i=I
However, our objective is usually to estimate the single-decrement proba­
bilities q�(d) and q�<w>, in this case from data that have been collected in a
double-decrement environment. To accomplish this, we could calculate estimates of q�<d) and q�<w) from the estimates
and
produced by (6.3la)
and (6.31 b), using one of the approximate relationships between single­
decrement and double-decrement probabilities presented in Section 5.3.
Alternatively, we could calculate estimates of <fx(d) and <fx(w) directly
(w) d"trectly m
(d) and
· terms
· s,-r,qx+r
tirom our samp Ie data by expressmg
,
s,-r,qx+r
,
of q�(d) and <Ix(w), under a chosen assumption, without using the preliminary
approximations (6.30a) and (6.30b). This approach normally results in equa.
.
f\l (d)
"' ( )
•
.
tions which must be solved numencally for qx and qx w
q�d)
2Jt>,
6.3.2 Uniform Distribution Assumptions
If the random events, death and withdrawal, are each assumed to have uni­
form distributions over (x,x+ 1) in the single-decrement context, we know
1(d) _ (u-r}·clx(d"J ,
0 '.S r < u '.S 1, and also that
from Section 3.5 that u-rqx+r - 1 _
r·q'/d"J
r(d) (d) - q',CdJ
(d) - q�(d")
)µx+u 1
,(d). Then Equation
JL.x+u - I - u·q'x(d) ' so that (1- u-rqx+r
-r·qx
(5.14a) becomes
Incomplete Data Samples: Moment Procedures
126
(d)
s-rqx+r
= q:
(d)
[cs-r)- f(s 2 -r 2). q:<w>]
(6.32a)
[ 1- r · q}d)] [ 1- r · /x(w)]
Similarly, Equation (5.14b) would lead to
(w)
s-rqx+r
-
-
q:(w) [<s-r) - 1(s 2 -, 2 ). q�(d)]
(6.32b)
(dJ] [ 1- r · q:
[ 1- r · q:
(w)]
Substitution of (6.32a) and (6.32b) into the basic moment equations (6.29a:)
and (6.29b), respectively, results in
E[Dx]
n
= I:
i=I
q:(d) [(s;-r;)- t(s?- r?) · q�(w)]
[ 1- r; · q:(d)] [ 1- r; · q:(w)]
= dx
(6.33a)
and
E[Wx] -
n
;
q:(w) [(s;-r;)- !(sl-,f) · ifx(d)l
[ 1- r; · q:<d)][ 1- r; · q:(w)]
-
Wx,
(6.33b)
The complexity of Equations (6.33a) and (6.33b), necessitating a numerical
solution, is obvious.
If Special Case A, in which r; = 0 and s; = 1 for all i prevails, then
estimation of q�(dJ and q:<w) is considerably simplified. We first note that
Equations (6.3 la) and (6.31b) simplify to
"(d)
- 4s.
(6.3 lc)
A(W)
- nx,
(6.31d)
qx
and
qx
nx
_Y.'t
Moment Estimation in a Double-Decrement Environment
127
respectively, under Special Case A. Then we can substitute these estimates
w)
w
of q;d) and qi ) for q}d) and ql themselves in Equations (5.20a) and
(S.20b), producing the estimates
(nx-twx+tdx) - V(nx-twx+idx) - 2nxdx
nx
2
(6.34a)
and
In Chapter 7 we will see that Estimators (6.34a) and (6.34b) are maximum
likelihood estimators.
Alternatively, we could simplify Equations (6.33a) and (6.33b) un­
der the Special Case A values of r; = 0 ands;= I and solve the resulting
equations directly for our estimates of q: (dJ and q: (w) given by Equations
(6.34a) and (6.34b). This alternative approach is pursued in the exercises.
ffEXAMPLE 6.61! I 00 persons enter the estimation interval (20,21] at exact
age 20. None are scheduled to exit before exact age 2 1, but one death and
two random withdrawals are observed before age 21. Assuming a uniform
distribution of both random events, estimate q;�d).
!SOLUTION! Using Equation (6.34a), we have lOO-t(2)+1(1) = 99.5, so
11.1 d)
99.5- ../(99.5)2 -200 - 0101015
D
that q20( 100
- •
•
The simplification of (6.33a) and (6.33b), and the resulting solution
11.t (d)
11.I (w)
.
of them for qx and qx , under Special Cases B and C, are left to the exercises.
6.3.3 Exponential Distribution Assumptions
If both death and withdrawal are assumed to have exponential distributions
(constant forces) over (x, x + l ), then, from Section 3 .5, we have the values
cd>
'<w> = e-<u-r)µ1"1 Equa'<dJ = e-<u-r)µ • and I - u-rqx+r
µ<d'J
x+u - µ<dJ , 1 - u-rqx+r
tion (5. l 4a) then becomes
128
Incomplete Data Samples: Moment Procedures
l(d) (d) d
l(W)
1
J µx+u U
J [ - u-rqx+r J [1 - u-rqx+r
r
s
( )
u-rXµ<dl + µM ) d
u
µ d J e-(
S
µ
(d )
µ<dl + µ <w)
[l _ e
-(s-r)(µ<d)+ µ<"'>)
]
(6.35 a)
Similarly, Equation (5.14b) becomes
(w
)
s-rqx+r =
+ µ(w) [ I - e-(s-r)(µ<d)+ µM)].
µ
(d)
(6.35b)
<w)
µ
Substitution of (6.35a) and (6.35b) into the basic moment Equations (6.29a)
and (6.29b), respectively, results in
(6.36a)
and
Wx.
(6.36b)
Again, the necessity of a numerical solution of these equations for µ(d) and
"'
-[J,< ) is obvious.
If Special Case A prevails, with r; = 0 and s; = l for all i, then we
(
can easily obtain our estimates of q:<d)and q: w)by substituting the estimates
w
w
of qid) and qi ), given by Equations (6.3lc) and (6.3ld) for qid) and qi )
themselves in Equations (5.28a) and (5.28b), producing,
= 1 _ [nx-dnxx -}Vx ]
d,/(d,+w,)
(6.37a)
d
w,/(d,+w,)
M(w)
qx
= 1- [nx - nxx -Wx ]
(6.37b)
Ai(d)
qx
and
In Chapter 7 we will see that Estimator (6.37a) and (6.37b) are maximum
likelihood estimators.
Moment Estimation in a Double-Decrement Environment
129
�LE 6.7lj Use the data of Example 6.6 to estimate q;�d), assuming
an exponential distribution for both death and withdrawal.
d
�OLUTION! Using Equation (6.37a), we have the values nx - x - wx = . 97
nx
d)
1 3
hen
1
.0101017.
1
D
d
T
-(.97)
=
=
=
q��
}.
x
x!
an d w
The simplification of (6.36a) and (6.36b), and the resulting solution
Al (d)
Al (w)
ofthem for qx and qx , under Special Cases B and C, are left to the exercises.
6.3.4 Hoem's Approach to Moment Estimation in a
Double-Decrement Environment
The mathematical difficulty inherent in the solution of (6.29a) and (6.29b)
.
.
/\,(w)
.&.
A,(d)
.
or exponent1a1 d"1stn"button, moti·c.
ior qx and qx , under e1t· h er the un11orm
vates us to find a different, simpler, approach to moment estimation in the
double-decrement environment. One such approach is the actuarial approach,
to be described in Section 6.4; another is Hoem's approach (see Hoem [36]),
which we present at this time.
For person i, who enters (x, x+ l] at age x+r;, Hoem contemplates a
theoretical age scheduled for withdrawal, say x+t;, in the same manner that
there is an age scheduled for exit due to the ending ofthe observation period,
which we have called x+s;. The difference is that x+s; is known when
person i enters (x, x+ 1 ], whereas x+t; is not known (except, according to
Hoem, to a Laplacian demon). For those persons with t; < s;, the values oft;
become known, a posteriori, as the sample experience unfolds and the
withdrawals are observed.
Using the exposure interpretation of the moment estimator devel­
oped in Section 6.2.3, we find exposure of(s;-r;) if person i survives to the
scheduled ending age, and exposure of (t1 -r;) if person i withdraws at age
x+t; < x+s; (i.e., survives to the "scheduled" withdrawal age). But what is
the exposure for those who die in (x, x+ 1 ]?
In Section 6.2.3 it was emphasized that the appropriate exposure for
a death was the scheduled exposure (s1 -r1 ), in a single-decrement environ­
ment. Now in our double-decrement environment, we will still use the sched­
uled exposure for a death, but the scheduled exposure is now (s;-r;) or
(t;-r;), whichever is less. In other words, if a person had an unknown
scheduled withdrawal time of!; < s;, and died before time I;, this scheduled
withdrawal time would never become known. This person should contribute
(t1 -r;), but we can never know what t1 is (or would have been).
130
Incomplete Data Samples: Moment Procedures
Hoem deals with this dilemma by simply assuming that no one who
died in (x, x+ l J had a ti value that was less than s;. In other words, al/
deaths contribute (s;-r;) to the exposure. Then the estimate of q:<d) is pro­
duced as the ratio of number of deaths to the exposure.
An example will make this procedure clear. The important point is
that all persons have a known value of S; at entry into (x, x+ l ], regardless of
the outcome for person i (survival, death or withdrawal). Each person's val­
ue oft; becomes known only if that person actually withdraws. Since t; can
never become known for a person who dies, si values are used for all deaths.
!!EXAMPLE 6.811 The basic data on a sample of four persons has been
analyzed to produce the following ordered pairs (r;, s;) with respect to the
estimation interval (x,x+ l]. In addition, it is known that one person died
and one person withdrew while under observation, at the times shown. Esti­
mate q:<d)_
Person
(r;, Si)
1
(0, .40)
(0, 1)
2
3
(.30, 1)
4
(.50, .80)
Time of
Time of
Death Withdrawal
-
.70
-
-
.75
!SOLUTION! Persons l and 3 contribute exposure of .40 and .70, respec­
tively. Person 4 contributes .25, the theory being that he was scheduled to
withdraw at age x+. 75, since, in fact, he did! Person 2 contributes 1, under
the assumption that she was not scheduled to withdraw before age x+ 1. The
·
Al(a)
}
0
total ex.posure ts .40+. 70+.25+ I = 2.35, so qx = 2_35 = .42553.
6.4 THE ACTUARIAL APPROACH TO MOMENT ESTfl1,fATION
Of the various approaches to the estimation of qx over the interval (x, x+ 1 J
described in this text, the actuarial approach was the first to be developed,
dating from the middle of the nineteenth century. There are two significant
implications of this early origin:
(1) The method predates the formal, scientific development of sta­
tistical estimation theory.
(2) The method predates any kind of mechanical or electronic cal­
culating equipment.
The Actuarial Approach to Moment Estimation
131
It will be easy to see the reflection of these two observations in the features
of the actuarial method. The method was developed primarily in an intuitive
manner, in the absence of a guiding statistical theory, and a key theoretical
concession was made in the interest of simpler record-keeping and calcula­
tion. The traditional actuarial approach remained in use well into the second
half of the twentieth century, even after the availability of both high-speed
data processing capabilities and modem statistical theory had made the pro­
cess nearly obsolete. Over the past ten years, however, we have seen mount­
ing evidence to suggest that this pioneering method of survival model
estimation should become, and is becoming, of historical significance only.
Therefore the presentation of the method here is in the nature of a historical
summary. The reader interested in a fuller history can pursue this through the
many references given here.
We will present the actuarial method assuming a double-decrement
environment. The simplification that results if the environment is single-dec­
rement will be easily seen.
6.4.1 The Concept ofActuarial Exposure
The key difference between the traditional actuarial approach to estimation
and the moment approaches presented thus far in this chapter is that the
actuarial approach has not generally made use of the concept of scheduled
exposure, as defined in Section 6.2.3. Rather it made use of a type of
observed exposure, which we will call actuarial exposure, to distinguish it
from the two other types of exposure (scheduled and exact) described earlier.
Recall that person i is scheduled to be an ender to the study at age zi ,
and this is known at entry to the study. Recall also that if x < zi < x+1, then
we say that person i is scheduled to exit (x,x+1] at age z; = x+si, where
0 < s; < 1. On the other hand, if z; 2: x+1, we say that person i is scheduled
to survive (x, x+1], which is the same as to say scheduled to exit at x+s;,
where si = 1.
Suppose person i enters (x,x+l] at age x+ri , 0 � r; < 1. Under the
actuarial method, if person i is an observed ender at x+s;, he contributes
exposure of (si-ri)- If he is an observed withdrawal at x + t;, where,
necessarily, I; � s;, he contributes exposure of (ti -ri)- But what if person i
is an observed death in (x,x+ l ]? We have seen that a proper moment
estimation procedure would have person i contribute exposure to age x+s;,
the scheduled exit age, and early in the historical development of the actu­
arial method this was intuitively recognized. However, the identification of
scheduled exit age, for a person who died before reaching it, required more
132
lncomplet"' Data Samples: Moment Procedures
extensive data analysis than the early manual procedures allowed. To resolve
the problem of an unknown x+s; for an observed death, it was simply
assumed that S; = 1 for all deaths.
The important point here is that the traditional actuarial approach to
handling the data, as thoroughly described by Batten [6] and Gershenson
[26], simply did not identify x+s; a priori. Rather x+s; became known a
posteriori only if person i was, in fact, an ender, just as Hoem's "scheduled"
withdrawal age x+t; becomes known, a posteriori, only if person i is an
observed withdrawal. Thus, in the actuarial approach, x+s; did not become
known for a death, so it was assumed to be x+ I, just as, in Hoem's ap­
proach, a death that might otherwise have been a withdrawal has an un­
known x+t;, and is therefore assumed to be x+s;.
Note that if, in fact, s; = l, then the actuarial method of giving
exposure to age x+ l is correct; it is only if a death has s; < 1 that the
method "over-credits" exposure to age x+ 1 rather than only to age x+s;.
!!EXAMPLE 6.9!! Three persons in a study sample all have the ordered pair
(r;,s;) = (0, .75). Person 1 remains in the sample to age x+.75, Person 2
withdraws at age x+. 50, and Person 3 dies at age x+.50. How much expo­
sure does each person contribute (a) under the actuarial approach; (b) under
Hoem's moment approach?
ISOLUTIONI (a) Person 1 contributes .75, from age x to age x+. 75; Person
2 contributes .50, from age x to age x+.50, where exposure ceases due to
withdrawal; Person 3 contributes 1.00, from age x to age x+ 1. (b) Person l
and Person 2 again contribute .75 and .50, respectively; Person 3 contributes
only to scheduled exit age, which is .75, under the moment approach.
D
It cannot be overemphasized that the historical reason for crediting
Person 3 in Example 6.9(a) with exposure to age x+ 1, rather than only to age
x+. 75, was that the scheduled exit age was not conveniently available infor­
mation in the actuarial approach to processing the sample data. Assuming
such scheduled exit age to be x+ l was simply a practical expedient.
Since age z; = x+s; is person i's age when the observation period
closes, then it follows that crediting exposure to age x+ 1, whenever s; < I,
means that person i is viewed as contributing exposure after the observation
period has closed. Ackland [ 1] referred to this as an "odd procedure" which
is "in strictness" incorrect, but nevertheless acceptable for reasons of expedi­
ency. Seal [67], [68] and [69] repeatedly criticized this facet of actuarial
moment estimation, especially later attempts to justify it by a statistical prin-
The Actuarial Approach to Moment Estimation
133
ciple. These attempts at statistical justification appear to have originated
with Cantelli [18] in 1914, and Balducci [5] a few years later. (Cantelli's
fa ulty argument is not presented here; the interested reader will find it in
[18).)
Cantelli's argument, although faulty, appeared to have been widely
accepted, at least in North America. Subsequent papers by Wolfenden (75],
Beers [9] and Marshall (54), and textbooks by Gershenson (32] and Batten
[8], in effect repeated his argument. But the preponderance of modern
statistical thought indicates that Cantelli's approach, although correct for
Special Cases A and B, is seriously flawed in the cases involving enders
within the estimation interval. (See, for example, Hoem (36) and Seal [68).)
In a single-decrement environment the possibility of withdrawals
es
not
exist. Then observed enders contribute exposure to their observed
do
ending ages, and observed deaths contribute exposure to age x+ 1, regardless
of a possibly earlier scheduled ending age.
6.4.2 Properties of the Actuarial Estimator
In the single-decrement environment, the actuarial estimators for Special
Cases A and B are the same as the moment estimators, so the discussion in
Section 6.2.5 regarding bias and variance is applicable to the actuarial esti­
mators as well.
In situations involving enders within (x,x+ l ], that is, enders at x+s;
wheres; < I, various investigations have shown under simulation, or other­
wise, that the actuarial estimator is negatively biased. It has also been
shown to be inconsistent. (See, for example, Breslow and Crowley [13], and
Broffitt [14].) In large�scale actuarial studies of insurance data, there will
always be a considerable number of enders for the observation period, and it
follows that there could therefore be a significant number in any estimation
interval (unless the study design places them at the interval boundary, which,
as we will see in Chapter 9, is frequently the case). The moment estimator
will always produce a larger estimate than will the actuarial estimator, so
a natural way to counteract the negative bias in the latter is thereby sug­
gested.
Finally it should be repeated that all estimators of the form £�;�;7:re,
which includes most of the ones in this chapter, are approximately binomial
proportions, so all are approximately unbiased with variance given by
p
Exposure
:r.·gx . This fact plays a crucial role in the next section.
t
134
Incomplete Data Sa mples: Moment Procedures
6.5 ESTIMATION OF S(x)
Thus far in this chapter we have concentrated on obtaining estimates of qx or
Px over the estimation interval (x,x+ I]. To estimate S(x) we will use the
approach introduced in Section 2.4.6, whereby
I\
S(x)
= Po · Pi · · · · · Px-1·
/\
/\
(6.38)
/\
To find the expected value and variance of S(x), we make two important assumptions:
I\
(1) We assume that each Pi , i = 0,1,- · ·,x-1, is approximately an
unbiased binomial proportion, with mean p; and variance 2f/1-,
q;
where n� is the exposure used to calculate
(and hence JJ;)­
Note that this exposure is being interpreted as an approximate
binomial sample.
(2) We assume that, conditional on the set {n;}, the p;'s are mutual­
ly independent. (In Chapter 9 we will explore a study design in
which the s are unconditionally independent.)
P;'
The important consequence of the first assumption is that, being unbiased,
E[p;ln;] = Pi· The importance of the second assumption is that the expected
value of a product of independent random variables is the product of their
expected values.
I\
6.5.l Expected Value ofS(x)
It is important to note that this expected value is conditional on the set ofex­
for
posures (approximate binomial samples) underlying the estimates of
I\
i = 0,1,... ,x- I. We will reflect this by using the notation E[S(x)l{n;}].
I\
With S(x) given by (6.38), we have
q;,
x-1
E[S(x)l{n;}] = E[p0 ·P 1 · ··· · Px-il{n;}] = IT E[p;l{n;}],
i=O
(6.39)
as a consequence of the independence assumption. Then from the assump­
tion of unbiasedness, we have
135
Estimation of S(x)
/\
E(S(x)f {n'} J = Po · p, · · · · · Px-1
= S(x).
(6.40)
/\
Thus we see that S(x) is unbiased under our pair of assumptions.
/\
6.5.2 Variance of S(x)
Again we note that this variance is conditional on the set of exposures, and
/\
we reflect this in the notation Var[S(x)I {n;}].
Proceeding from first principles,
2
Var[S(x)l{n;}] = E[S(x) 1 {n;}] - (E[S(x)I {n;}]
/\2
/\2
/\2
= E[po · Pi · · · · · Px-1 I {nI;}]
)
2
2
- (E[po ·p1 · ··· · Px-1l{n;}J)
(6.41)
I\
by the definition of S(x) given by (6.38). The independence assumption then
allows us to write (6.41) as
x-1
I\
Var [S(x)l{n;}] = TI E[p; l{n;}] 2
i=O
Now for each
·
(6.42)
P;,
so that
But since
iS
P; is taken to be an unbiased binomial proportion, the first moment
E[p;I {n;}J = Pi,
(6.43)
and the second moment is
(6.44)
Substituting (6.43) and (6.44) into (6.42), we have
Incomplete Data Samples: Moment Procedures
136
II
(Pi)2 · (� +
Var(s(x)J {n'.}) =
P1n1
1=0
=
(x-1
TI p;
r=O
)2
= [S(x)]2 ·
since S(x) = po· Pi · ·· · · Px-1
x-1 (
1)- (Ylp;
)
1=0
q1
2
(x-1
. 1TI
-.- + l ) -- 1=0
TI p;
=0
P1n;
[fl (
)2
I
;
q, +
1=0 P1n;
1) - 1],
(6.45)
= IT p;.
r-1
The exact expression for the conditional variance of S(x), given by
(6.45), is sometimes approximated in the following manner. Expanding the
product of the binomial terms, we obtain
i=O
Yii=O \I pq;n;,; + 1) = (1 + p::,) (1 + p��} (1 +
= I+�+ __!]J_ + ··· +
pono
p,n;
p,���J
qx-l
+ (higher
Px-,n:_,
order terms).
Since these second and higher order terms are quite small, we ignore them,
(
)
and approximate 1}'. �+ l by 1 +
x-1
�
�-Thus (6.45) is approx�mated by
Var [S(x)I {n;}] � [S(x)] ·
2
q;, ·
L
i O Pi l;
x-1
=
l
(6.46)
This formula for the conditional variance of S(x) is known as Greenwood's
formula (see Greenwood [34]). Since the terms which we ignored were all
positive, Greenwood's approximation understates the true value of this variance.
Greenwood's approximation to the variance can also be derived by
the method of statistical differentials, described in Section 2.6. We note that
S(x) is a function of the x random variables p0, ... ,Px - i, specifically, the
product of them. Thus Equation (2.79), extended from two to x random
I\
variables, can be used to approximate the variance of S(x). Because the p;'s
are conditionally independent, then all covariance terms in (2.79) are zero.
The general term in (2.79) is the derivative of S(x) with respect to /J;,
evaluated at 1 = PJ (i.e., the mean of 1), for j = 0,1, ... ,x-1, squared,
times Var(p;). Thus
p
p
137
Estimation of S(x)
Var[S(x)l{ni}) =
/\
I
x-l
2
L ( ffv. S(x)I" .) · Var(p;),
1=0
{)
/\
'P,
/\
P1=P1
(6.47)
The simplification of(6.47) into (6.46) is left as an exercise.
[j:XAMPLE 6.1011 In a study using a moment estimation approach, we ob­
serve the following numbers of deaths in estimation interval (x,x+ l], based
on the given exposure amounts.
Interval
(0,1]
( l,2]
(2,3]
(3,4]
(4,5]
Deaths
IO
8
7
7
8
Exposure
1040
1120
1191
1307
1612
Estimate S(5) and Var[S(5)l{n�}] from this sample data.
/\
[[OLUTION! From the sample data we obtain the following values:
i
0
1
2
3
4
n�l
1040
1120
1191
1307
1612
Then S(5) = Po
/\
/\
/\
p;
q;
10/1040 1030/1040
8/1120 1112/1120
7/1191 1184/1191
7/1307 1300/1307
8/1612 1604/1612
/\
I
p;n;
1030
1112
1184
1300
1604
".//\
I
q; P;n;
9.3J5 x 10-6
6.423 x 10-6
4.964 x 10-6
4.120 x 10-6
3.094 x 10-6
·p 1 ·p2 ·p3 ·p4 = .96747. The variance is estimated as
(.96747) 2 [(l .000009335)(1.000006423)(1.000004964)
(1.000004120)(1.000003094)- J ),
which is(.96747) 2 · (27.93631 x 10-6) = .000026148336. Ifthe Greenwood
approximation to (6.45) is used, the approximate estimated variance is
(.96747) 2 · [(9.335+ 6.423+4.964+ 4.120 +3.094) x 10-6], which evaluates
to (.96747)2 · (27.936 x 10-6) = .000026148046. This gives us a good pic­
ture ofthe small understatement inherent in Greenwood's approximation. 0
138
Incomplete Data Samples: Moment Procedures
6.6 SUMMARY
Most of the estimators for qx (or for q:< > in double-decrement notation)
derived in this chapter are in the form of a ratio of observed deaths over a
measure of exposure from which those deaths arose. In addition we noted
that an estimator for mx (or m� (d) in double-decrement notation) is also in
that ratio form, and this quantity is an estimator for µ, the constant force
over (x, x+ l) as well. Therefore, much of this chapter is summarized accord­
ing to the methods by which exposure may be calculated.
Recall that the fh life in the sample begins contributing exposure to
the estimation interval (x,x+ l] at age x+ri, 0 ::; r; < 1, and is scheduled to
cease contributing exposure to that interval at age x + s;, 0 < s; ::; l , either by
reaching the end of the interval, in which cases; = I, or by the intervention of
the end of the observation period, in which cases; < 1. It is possible that the
fh life dies at age x+t;, where, necessarily, !; s;, or that it withdraws at age
x+t;, t; s;. If neither random event occurs, then person i is called an
observed ender at x + s;, if s1 < I, or an interval survivor at x+ 1, if s1 = I.
For each of the estimators of the general form /;;i:'::re , Table 6.1
d
s
s
summarizes the determination of exposure.
TABLE 6.1
Type of
Calculation of Exposure
Estimator
Exposure
Exact s; - r; for observed enders and survivors
m
/\x , or (constant) µ
I; - r; for observed deaths and withdrawals
I\
Scheduled s; - r; for observed enders and survivors
t; - r; for observed withdrawals
moment approach
s; - r; for observed deaths
qx , by Hoem's
I fix , by the actuarial Actuarial s; - r; for observed enders and survivors
f; - r; for observed withdrawals
l - ri for observed deaths
i approach
L
In the single decrement case, the moment estimator i s derived using
the general approximation b-aqx+a � (b-a) · qx. Under Special Case B this
approximation is the same as the hyperbolic assumption, and under Special
Case C it is the same as the linear assumption.
The actuarial estimator was derived from a faulty moment equation
which naturally involved the function 1-uCfx+u· Thus it \Vas natural for it to
Summary
139
use the hyperbolic assumption indentity 1-u qx+u = (1-u) · qx , commonly
called the Balducci hypothesis in the actuarial literature. This matter is now
of historical significance only.
Some investigation of the bias and the variance of moment and actu­
arial estimators was presented.
A different moment approach in the double-decrement environment
more
complex than either the actuarial or the Hoem approach is given
is
t
tha
by Equations (6.33a) and (6.33b) under uniform distributions, and by Equa­
tions (6.36a) and (6.36b) under exponential distributions. If a computer rou­
tine to solve these equations is readily available, they are an excellent ap­
proach to the estimation of q� (d) and q�(w)_ In particular, if Special Case A
•
Al(d)
Af W)
fior qx and qx( are ava1·1able.
•
preva1·1s, convement so1ut10ns
A different moment approach for Special Case C in the single-decre­
ment environment was presented in Section 6.2.6. This approach, called the
Schwartz-Lazar estimator, is used in clinical studies where the single-decre­
ment Special Case C frequently arises.
Finally, as an overall observation on moment procedures, there seems
to be no totally satisfactory way to deal with the presence of withdrawals in
the estimation of q�(d)_ The actuarial approach made use of a questionable
moment relationship and the illogical hyperbolic assumption with its
decreasing force of mortality over (x,x+1). Haem's approach requires the
assumption of a mystical "scheduled withdrawal age," and the convenient
assumption that no deaths would otherwise have been withdrawals. The
more analytical moment approach requires a numerical solution of the pair
of moment equations.
In light of these difficulties and since random withdrawals occur fre­
quently in actuarial and other mortality estimation problems, moment ap­
proaches are not generally recommended. We will see in the next chapter
that maximum likelihood procedures, although still requiring simplifying
assumptions at times, are generally superior to moment approaches.
6. 7 EXERCISES
6.1 Introduction
6.2 Moment Estimation in a Single-Decrement Environment
6-1. Find an expression for s-rq.t+r, in terms of qx , under the linear as­
sumption for fx+s·
140
Incomplete Data Samples: Moment Procedures
6-2.
Find an expression for s-rqx+r, in terms of q_n under the hyperbolic
assumption for fx+s ·
6-3. Consider the approximation s -rq.Y+r � (s - r) · qx.
(a) Show that this implies a linear f.x+s ifr = 0.
(b) Show that this implies a hyperbolic f.x+s ifs= l.
(c) Show that there exists no decreasing f.x+s function for which
s-rqx+r = (s-r) · qx for all r and s. (For this reason, this rela­
tionship does not represent a mortality distribution assumption
in the same sense as is true for the linear, exponential, and
hyperbolic assumptions.)
6-4. Show that the expression for s-rqx+r in-terms of qx , under each of the
linear, exponential and hyperbolic assumptions for f.x+s, reduces to
(s-r) · qx if quadratic and higher powers of qx are ignored.
6-5. Suppose, of the nx persons who contribute to (x,x+ I], kx of them have
r; = r, a constant, and the remaining (nx -kr ) have r; = 0. Suppose all
nx haves; = I. Derive the moment estimator of qx.
6-6. Over the estimation interval (x, x+ 1 ], the central rate of mortality is
estimated as fnx = . 015. There were two observed deaths in the in­
terval, one at age x+ and one at age x+ and both deaths had s; = I.
Find the moment estimate of qx.
!
f,
6-7. Show that Estimator (6.7) for Special Case A is unbiased.
6-8. (a)
(b)
Show that Estimator (6.13) for Special Case C is unbiased under
the linear assumption for f.x+s·
If the force of mortality were actually constant over (x,x+ !),
what is the bias in Estimator (6.13)?
6-9. Find exact expressions for the variances of each of the following esti­
mators:
(a) Estimator (6 .7)
(b) Estimator (6. l 0 )
(c) Estimator (6.13)
Exercises 6-10 through 6-12 relate to the Schwartz-Lazar moment estimator
described in Section 6.2.6.
141
Exercises
6-10. Suppose all Cx persons have a common scheduled ending age of x+s.
Find an expression for o: under each of the following assumptions for
£.,+s:
(a) Uniform
(b) Exponential
(c) Hyperbolic
6-11. Now suppose the ex scheduled enders have scheduled ending ages that
are distributed over (x,x+ 1] with PDF g(u) and CDF G(u).
1
(t)Jdt
(a) Show that, in general, o: = f0 ,P;,i:x�dl-G
o 1Pxµx+1dt
(b) Find the expression for o: if g(u) = l (i.e., if Uhas a uniform
distribution over (x, x+ I)), and if the mortality assumption is a
linear £x+s ·
(c) Repeat part (b) assuming that Uhas a uniform distribution, but
the mortality assumption is an exponential fx+s ·
6-12. Find an estimator of qx by eliminating o: from Equations (6.25) and
(6.26). (This is the reduced sample estimator of Drolette [23].)
6.3 Moment Estimation in a Double-Decrement Environment
w
6-13. A traditional formula used for approximating q:(d) from q�d) and q� ) is
(d)
w)· Use this relationship, along with (6.3la) and (6.3lb)
I -lq}
to estimate q}d).
q:(d) ;=:;j
6-14. Assuming Special Case A prevails, show that Equations (6.33a) and
(6.33b) lead to Estimators (6.34a) and (6.34b).
6-15. Assuming grouped Special Case B prevails, with r; = r for all i, solve
.
�M
�M
Equations (6.33a) and (6.33b) for (a) qx , and (b) qx
6-16. Assuming grouped Special Case C prevails, with s; = s for all i, solve
.
", d)
",(w)
Equat10ns (6.33a) and (6.33b) for (a) qx( , and (b) qx
6-17. Assuming. Special Case A prevails, show that Equations (6.36a) and
(6.36b) lead to Estimators (6.37a) and (36.7b).
Incomplete Data Samples: Moment Procedures
142
6-18. Assuming grouped Special Case B prevails, with ri = r for all i, solve
.
N�
NM
Equations (6.36a) and (6.36b) for (a) qx , and (b) qx · .
6-19. Assuming grouped Special Case C prevails, with s; = s for all i, solve
.
/\_t(d)
/\_t(W)
Equations (6.36a) and (6.36b) for (a) qx , and (b) qx · .
6-20. Suppose 100 persons come under observation at age x, and an addi­
tional 30 persons come under observation at age x+t. There are 2
deaths in (x,x+l]. Solve Equation (6.16) fort, under each of the
hyperbolic, linear, and exponential assumptions.
6-21. A study sample has 10,000 persons at age x. There are l 000, 100, and
10 intermediate entrants at ages x+i, x+f, and x+f, respectively, and
3 730 deaths in (x, x+ 1]. If µx+t is constant for O < t < 1, find the
moment estimator of qx .
6.4 The Actuarial Approach to Moment Estimation
6-22. A group of 100 insureds is observed for a period of one year. During
the year there are two deaths and six withdrawals. Deaths and with­
drawals occur ccnlraUy in the year. Estimate the annual central rate of
withdrawal, m(w) , and the annual probability of withdrawal, qM, by the
actuarial approach.
6-23. A study over the period January 1, 1990 to December 31, 1991 yields
the following information:
Individual Date of Entry Date of Death Date of Withdrawal
v Janua1y 1, 1990
September 30, 1991
w January 1, 1991
x January 1, 1990 June 30, 1991 March °31, l 990
y
January l, 1990
z
??
December 31, 1990
The estimation interval is one year. The central rate mo is estimated by
fno = .33. Using the traditional actuarial estimator, estimate q0.
Exercises
143
6-24. Ten individuals were observed for all or part of the two-year period
following attainment of age l00. Five persons survived to age 102.
One person withdrew at .75 of the two-year period. Four persons
died, at du rations .20, .55, .70 and .80 of the two-year period.
Estimate 2m1006-25. Suppose n lives come under observation at age x+a, and will leave
observation at age x+b, 0 S a < b I, unless death intervenes. Let d
be the number of deaths that do intervene.
(a) Verify the moment equation n · b-aq.Ha = d.
(b) Show that b-aqx+a = 1-aqx+a - b-aPx+a · 1-bq.Hb·
(c) Substitute (b) into (a), make the hyperbolic assumption substit­
ution for 1-aqx+a and 1-bqx+b, and reach the moment equation
n(l-a) · qx - n · b-aPx+a · (1-b) · qx = d.
(d) Note that n · b-aPx+a is the expected number of survivors at age
x+b. Substitute for it (n-d), the actual number of survivors in
the sample, and solve the resulting equation for the traditional
actuarial estimator. (This derivation of the actuarial estimator is
given by Broffitt and Klugman [ 15].)
s
6-26. In a Special Case C study, there are ex scheduled enders at age x+s out
of nx alive at age x, of which ex survive to age x+s and are thus
observed enders.
(a) What is the expected number of deaths to occur in (x,x+ 1) out
of the nx beginners?
(b) What is the expected number of deaths to occur in (x+s, x+ l] to
persons who have left the sample
(i) conditional on being alive at age x?
(ii) conditional on the number actually leaving the sample at
agex+s?
(c) Equate (a) - (b) to the number of observed deaths. Solve the
equation using the linear assumption under (b), (i) for the moment
estimator, and using the hyperbolic assumption under (b), (ii) for
the actuarial estimator.
(d) Compare our alternatives (b), (i) and (b), (ii) with the derivation in
Exercise 6-25.
Incomplete Data Samples: Moment Procedures
144
6-27. Consider the Special Case C study of Exercise 6-26. Let Dn be the
random variab1e for all deaths in (x, x+ l] out of the nx beginners, and
let De be the random variable for the (unobservable) deaths in
(x+s, x+ 1] out of those who terminate observation at age x+s. Then
Dx = Dn -Dc , and in Exercise 6-26 we found
which led to Estimator (6.19).
(a) ls it true that Var(Dx ) = Var (Dn ) + Var(Dc )? Why?
(b) Use the fact that Cov(Dn ,Dc ) = c(l-qx )(qx - s qx ) to derive
Var(Dx )-
t
6-28. Show that, in a Special Case C study,
by the traditional actuarial ap­
proach cannot exceed unity, whereas qx by the moment approach can.
6.5 Estimation of S(x)
6-29. Evaluate Equation (6.47) to reach Greenwood's formula given by (6.46).
6-30. Explain by general reasoning why Equation (2.79) produces the same
approximation to Var [S(x)!{n;}] as does Greenwood's approximation.
CHAPTEJR 7
TABULAR SURVIVAL MODELS ESTIMATED
FROM INCOMPLETE DATA SAMPLES
MAXIMUM LIKELIHOOD PROCEDURES
7.1 INTRODUCTION
In this chapter we consider maximum likelihood estimation (MLE) as an
alternative to the actuarial and other moment estimators developed in Chap­
ter 6. It is assumed that the reader has a basic familiarity with MLE from a
study of statistical estimation in general. Here we will be applying MLE
theory specifically to estimate qx over the basic interval (x, x+ 1 ], considering
both single-decrement and double-decrement environments. As in Chapter 6,
we will see that the single decrement environment is the more convenient
one within which to estimate qx.
In our moment approaches of Chapter 6 we were equating the expec­
ted number of deaths in (x, x+ l] to the number actually observed. This
meant that we needed only the observed number of deaths in (x, x+ I], which
we called dx , and not the age location of those deaths within(x,x+l]. Recall
that we did make use of the specific ages at which a person began observa­
tion and was scheduled to end observation, or was observed to withdraw, but
we never used the precise ages at death of the d_T group to estimate qx .
In MLE, the precise age at death is information that can be incor­
porated into the estimation procedure if it is available. Thus we distinguish
two important subdivisions of MLE: the situation in which precise age at
death is used will be referred to as afull data situation; that in which only
the number of deaths in the interval (x, x+ 1] is used will be referred to as a
partial data situation. Note that the very nature of moment estimation calls
for partial data only on the deaths, whereas maximum likelihood estimation
can be based on either partial data or full data.
Thus we will find that an MLE problem will begin with a recog­
nition of the available data, including the question of whether we have full
data or partial data on the deaths. This information will lead us to the
likelihood function. We will then need to make certain simplifying assump­
tions, such as mortality distribution assumptions, in order to solve the likeli­
hood equation for qx .
146
Incomplete Data Samples: Maximum Likelihood Procedures
The concept of scheduled exit age, x+si , for person i with respect to
the estimation interval (x, x+ l J will not play a major role in this chapter.
Maximum likelihood estimation of interval mortality probabilities is
discussed in many other texts on survival analysis and survival model esti­
mation. The work of Broffitt [ 14 J is a particularly good one, and parts of this
chapter are based on that work.
7.2 SINGLE-DECREMENT ENVIRONMENT, SPECIAL CASE A
We wish to start with this case as a way of obtaining familiarity with the
maximum likelihood approach in the simplest possible situation. We can
then deal with more complex situations knowing that the basic theory is
comfortably understood.
7.2.J Partial Data
As defined in Section 6.2.2, this situation simply states that, of nx lives
exactly age x, dx of them die in (x, x+ 1], and nx - dx survive to age x+ 1.
We recognize this as a binomial model, so the likelihood is simply the bi­
nomial probability of obtaining the sample result actually obtained. That is,
nx !
( d, (1 q )n,-d,.
L(qx I flx,d)
x
= dx l(nx dx )! qx )
- x
-
(7.1)
One of the basic properties of MLE is that any multiplicative con­
stants can be ignored, and the same estimate of qx will still result. When this
is done, the likelihood is no longer the probability of the sample per se, but
rather is proportional to it. Thus many writers prefer to write
(7.2)
where ex is read "is proportional to." We wish to take the point of view
that it is just as reasonable to call the right side of (7.2) the likelihood itself,
as to call it something to which the likelihood is proportional. Thus we
would write simply
(7.3)
The notation L(qx lnx ,dx ) reminds us that the likelihood is a function
of the unknown qx , and that nx and dx are given values, namely those ob­
served in the sample upon which our estimate of qx is to be based. When
Single-Decrement Environment, Special Case A
147
there is no doubt as to the unknown and the given values, we will simply use
L instead of L(qx lnx ,dx ). Finally, for convenience we will frequently suppress
the subscript x. Thus we will write the likelihood for the Special Case A par­
tial data situation as
q,
(7.4)
The idea now is to find the value of q, called which maximizes
Equation (7.4). Formally, if exists, such that L(q) � L(q) for all other q,
then is the maximum likelihood estimator (MLE) of q.
To find by the calculus technique of setting �� = 0 would require
q
q
q
an application of the product rule for differentiating (7.4 ). Although this is
not difficult in this case, we will see that likelihood functions can become
much more complex, and several applications of the product rule might be
required. To avoid this, we take the natural log of L before differentiating.
That is, we solve the equation � InL = 0 for
The same value of will
q.
q
result from solving this equation as from solving �� = 0, since the natural
log is a one-to-one monotonic transformation.
Thus we define the log-likelihood, £, to be In L, obtaining, in this
case
£
Then
= lnL = d· lnq+(n-d)· ln(l-q).
(7.5)
d£ _ d. _ n - d _ o
dq - q
1 -q -
(7.6)
is the likelihood equation which easily produces
dx
I\
qx - nx '
(7.7)
which is the same as Estimator (6. 7), already noted in Section 6.2.2 to be an
MLE.
!!EXAMPLE 7.tll Show that the same qx will result for Special Case A par­
tial data even if the log transformation is not used.
ISOLUTIONI We differentiate (7.4) to obtain
�� = l[ - (n-d)(l- q)11-d-1] + (l- q)"-d[dqd-1] = 0,
so d · qd-l (1- q)"-d
is the same as (7.6).
= (n-d)q (l- q)"-d-l, or d(l-q) = (n-d)q, which
d
D
148
Incomplete Data Samples: Maximum Likelihood Procedures
!!EXAMPLE 7.1]j Show that 'cir. defined by Estimator (7. 7) maximizes,
rather than minimizes, (7.5).
d
n-d
s· df. d n-d h d 2f.
ISOLUTION mce dq = q - l-q' t en dq2 = - q2 - (l-q)2- Since
d, (n-d), q2 and (l-q)2 are all positive, then the second derivative is nega­
tive, so qx maximizes In L.
O
7.2.2 Full Data
Now we assume that we have the precise age at death for each of the dx
deaths in the interval. Since this age is different for each death, we consi der
them individually, and take the product of each death's contribution to the
likelihood function.
The likelihood for the i'h death is give::n by the:: probability density
function (PDF) for death at that particular age, given alive at age x. That is,
for death at age x1,
f(x,- X > x ) _ f(x;} _ S(x,) · A(x;)
L-1 _
I
- S(x) S(x)
(7.8)
is the contribution to L of the jlh death. If we let S; = X; - x be the time of
the i'h death within (x,x+l], where O < s; :S I, then we have
(
)
L _ S x+s;) · A (x+s; _
(7.9)
- s,Px µx+s;,
,
S(x)
in standard actuarial notation. The contribution to L for all deaths combined
IS
d
IT s;Px µx+s; ,
i=I
(7.10)
which is commonly written as fl s,Px µx+s,, and read as "multiplied over all
D
deaths."
Of course the nx - dx survivors contribute (Px Y1,-d, = ( 1-qx)"'-d,
to L, so we have the total likelihood
L __
er q )",-d, ITv s,P /L_,-+s,
- .t
•
x
(7.11)
for our Special Case A full data situation.
To solve (7.11) for 'cfx it is necessary to make a distribution assump­
tion which will express sPx µx+s, in terms of qx. We will consider two such
assumptions.
Single-Decrement Environment, Special Case A
149
Under the linear, or uniform, assumption, we find from (3.58) that
for all S;, 0 < s; :S 1, sPxµx+s; = qx . Then the contribution to L would be qx
for all dx deaths, so the tenn TisPxµx+s, in (7.1I) becomes (qx /', and (7.1 I)
D
d
becomes qd ·(I-qy- , which is (7.4). Thus (7.11) also leads to Estimator
(7.7). That is, full data Special Case A, evaluated under the uniform
distribution assumption,produces the same estimator as does the partial data
Special Case A, which requires no assumption for its evaluation.
Under the exponential assumption, we recall thatµx+s, is a constant,
s
namelyµ= - lnPx,and,from(3.64),sPx = (Px Y; = e-µ ',so that(7. l l )
becomes
(Px r-d, . IT (PxY' .µ
L
v
= µ' · exp [ - µ(n-d) - µ - � ,,]-
(7.12)
s,J.
(7.13)
Then
f = lnL = d - tn,, - µ [(n-d) + �
and
i- [(n-d)+ �S;] - Q,
(7.14)
which is easily solved for
µ
d
(n-d) + 2:s;
v
(7.15)
The denominator of (7.15) is the exact exposure mentioned earlier in the
text. Thus(7.15) estimates mx , which is the same asµ under the exponential
(constant force) assumption. In turn,
(7.I6)
is the MLE of qx . Since qx andµ have a one-to-one correspondence, we can
write (7.11) in terms of µ, solve for the MLE and then obtain the MLE
by (7.16). Alternatively, and more mathematically complex, we could write
(7.11) in terms of qx and proceed to the same qx . The demonstration of this
is left as Exercise 7.4.
µ,
t
150
Incomplete Data Samples: Maximum Likelihood Procedures
7.2.3 Summary ofSpecial Case A
We have derived two Special Case A MLE's in this section: (7.15) for the
full data, exponential distribution, and (7.7) for both the full data, uniform
distribution and the partial data situations. We have shown the mathematics
of these derivations in considerable detail to make the reader feel comfor­
table with the MLE approach. In future derivations we will not need to show
each step in such detail.
We now want to show an alternate way to write the likelihood func­
tion for the full data model. Let us define t; to be the time at which the i1h
person in nx ceases observation, either by reaching age x+ l (so that t; = \
and person i is a survivor), or by death (so that t; ::; 1 and person i is a
death). Define 8; to be an indicator variable where
_
b; - {
l if person i dies in (x, x+ 1J
O otherwise
(7. \ 7)
Then the contribution to L by the fh person in nx is
(7.18)
The structure of (7.18) shows that L; = 1,Px µx+i, if the l" person is a
death (since D; = l ), and L; = 1,Px if the i1h person is a survivor (since
15; = 0). Of course t; = 1 for all i for which D; = 0 in Special Case A, so it
might not- seem as if this approach to writing L; has any advantage in this
case. But we are introducing it at this time, since we will use it extensively
to write L; for more complex cases where its advantage will be more easily
seen.
l!EXAMPLE:::::zJJ] Of l 00 lives exactly age x, 2 are observed to die in the
estimation interval (x, x+ 1] and 98 survive to age x+ 1. The deaths occur at
precise ages x+.20 and x+.60. Estimate qx assuming the force of mortality
is constant over (x,x+ \ ].
-h
!SOLUTION! From (7.15), fl= 98+.io+.60 = 2 is the MLE ofµ when
0
/J. is constant. Then qx = l - e-51247 = .020039.
Single-Decrement Environment, General Case
151
7.3 SINGLE-DECREMENT ENVIRONMENT, GENERAL CASE
We will now use the ideas developed in Section 7.2.3 to develop a general
form for the likelihood function in a single-decrement environment, and then
show how Special Cases B and C, defined in Section 6.2.2, are contained in
the general fonn. Of course Special Case A is contained in the general form
as well.
7.3.1 General Form for Full Data
We recall that nx is the total number of persons in the sample who contribute
to the estimation interval (x,x+ 1], and x+r; is the age at which the i 1h person
enters (x,x+ 1], 0 ::;; r; < 1. Let x+t; be the age at which the i1h person
leaves (x,x+ 1], 0 < I; ::;; 1, whether as a scheduled, and observed, ender, as
an interval survivor, or as a death. Let O; be the indicator variable for person
i defined by (7.17). Then if!; = 1 and O; = 0, person i is a survivor; if I; < 1
and O; = 0, person i is an observed ender; if I; ::;; l and 8; = 1, person i is a
death.
As of age x+r;, person i is scheduled to leave observation at some
age not later than x+ I, but the possibility of prior death implies that the age
at which observation actually does cease is a random result. Thus t; is the
realization of a random variable T;. On the other hand, r;, the time of entry,
is not a random variable.
We can now see that if r; = 0 for all i, we have either Special Case A
or Special Case C. Similarly, if I; = I for all i for which O; = 0 (i.e., for all i
who do not die), we have either Special Case A or Special Case B. This is
summarized in Table 7.1.
TABLE7.1
t; = l for all i
for which 15; = 0
t; < I for some i
for which b; = 0
r; = 0 for all i
Special Case A
Special Case C
r; > 0 for some i
Special Case B
General Case
The general form of the contribution to L by person i is
L;
= t1 -r1Px+r,(µx+1)61
,
(7 .19)
152
Incomplete Data Samples: Maximum Likelihood Procedures
since, if O; = 0, the iikelihood is merely the probability of survival from x+r;
to x+t;, and if D; = l, it is the density function for death at x+t; given alive
at x-t-r;. The overall likelihood is
L =
IT t,-r,Px+r, (µx+1,)°' ·
(7.20)
i=l
To evaluate (7.20) we must make a distribution assumption. We will again
consider two cases.
7.3.2 Full Data, Exponential Distribution
Under the exponential (constant force) assumption, we find
L = (µt. IT e-(t,-r,)µ.,
(7.21)
i=I
so
II
f = lnl - d· Inµ-µ· I:(t;-r;),
and it is e�sy to see that
(7.22)
i=I
;!
O produces
µ
(723)
n
EU;-r;)
i=I
Estimator (7.23) is of the same form as Estimator (7.15), namely the ratio of
dx to the exact exposure of the sample, which is the sample central rate.
(This will be the result for all cases of full data evaluated under the expo­
nential assumption.) Thus (7.23) is the general full data MLE, and Special
Cases A, B, and C are all contained within it.
7.3.3 .Full Data, Uniform Distribution
To evaluate (7.20) under the uniform distribution assumption, we first note
that
l - l; · qx
"f= r;. q_�'
(7.24)
Single-Decrement Environment, General Case
and
µ_,+r, =
153
(7.25)
1 - I; · qx
When (7.24) and (7.25) are substituted into (7.20), we find that (7.20) will
contain qx for each death, (1- I;· qx) for each interval survivor (t, = 1) and
observed ender (t; < I), and (I - r, ·qx )- 1 for everyone. Thus (7 .20) becomes
n
L = (qx l. ITO-r,·qx r1. IT (I-l;·qx).
r=I
S&E
(7.26)
d
Then d In L = 0 produces
q
0.
(7.27)
In general, Equation (7.27) must be solved for qx by iteration. We
note that if r; were different for all nx persons, and if I; were different for all
nx- dx non-deaths, then there would be I + n + (n-d) = 2n - d + I dis­
tinct denominators in (7.27), producing an equation in qx of degree 2n-d
Of course we expect many r; = 0 and many t; = I, so the degree of the
polynomial will not be that big.
Since (7.27) is a polynomial in qx , we can see that a convenient solu­
tion is possible if that polynomial is of degree one or two. How can this be
accomplished?
If r; = 0 for all nx persons in the sample, and if I; = l for all nx -dx
non-deaths, then we have Special Case A, and (7.27) becomes (7.6), which
has two terms, so it solves linearly for Estimator (7.7).
If I; = 1 for all nx -dx non-deaths, r; = 0 for bx of the nx persons in
the sample, and r; > 0 for the remaining kx = nx -bx , then we have Special
Case B. If r; = r, a constant, for all kx persons, then we have the grouped
Special Case B defined in Section 6.2.4. Then (7.27) becomes
1
d_
( r ) - (n-d) (1 q)
q + k U-r·q
= 0,
(7.28)
0,
(7.29)
from which we obtain the quadratic equation
r(n-k)q - (n-rk + rd)q + d
2
154
Incomplete Data Samples: Maximum Likelihood Procedures
which leads to the result
b - Jb2 - 4r(nx-kx)dx
2r(nx-kx)
(7.30)
where b = nx - rkx + rdx. The negative radical is used in (7.30) because
the positive radical would result in qx > 1. This is also true for Equations
(7.32) and (7.70).
Similarly, for grouped Special Case C with common ending age x+s
for all scheduled enders, (7.27) will again become quadratic. This time
ri = 0 for all nx persons in the sample, ti = s for all ex observed enders, and
ti = 1 for the nx - ex -dx interval survivors. Then (7.27) becomes
s
-1 )
ef..
q - e(1-s · q ) - (n-e-d)(1-q
= O,
(7.31)
which leads to the result
(7.32)
where b = nx - ( 1-s)ex + sdx. Note that (7.32) is similar to (7.30), with r
replaced bys, k replaced by -e, and n-k replaced by n.
To complete our analysis of (7.27), we note that in the general case,
even if all kx entrants with r; > 0 are at a common entry age x+r and all ex
observed enders are at a common exit age x+s, the presence of an nx -kx
group at ri = 0 and an nx- ex - dx group at f; = 1 as well, means that there
are four distinct denominators in (7.27), which produces a cubic equation in
qx.
j[EXAMPLE 7.411 (a) For the estimation interval (x,x+l], suppose bx= 5
persons enter at age x, k.x = 3 persons enter at age x+.25, ex = 2 persons are
observed enders at age x+. 50, one death occurs at age x-t-.25, and a second
death occurs at age x+. 75. Calculate cix under the exponential assumption.
(b) Let the kx = 3 entrants occur at age x rather than age x+.25, and calcu·
late qx under the uniform assumption.
!SOLUTION)
(a) Since the ordered pair (r;, !;) is not given for each person, we find the
exact exposure of the sample by crediting each person with exposure
to age x+ 1, and then deducting the portion of credited exposure not
Single-Decrement Environment, General Case
(b)
155
realized. In this manner the exact exposure of the sample is found to
be 5+75(3-50(2) - .75 - .25 = 5.25. Under full data, exponential,
8
8 21
µ = faa:::;�sure = 5.�5 = 2 1. Then cJx = 1 - e- 1 = .31678.
We now have Special Case C, with n = 8, d = 2, and e = 2 \Vith end-
ers at age x+ 2l . Then (7.32) gives qx
I\
8-
J64 -32
8
= .29289.
0
7.3.4 Partial Data, Special Case C
Suppose all nx persons who enter (x,x+l] do so at exact age x (i.e., r; = 0),
and suppose that ex of them have a common, or average, scheduled ending
age of x+s, 0 < s < 1. The remaining nx - Cx are scheduled to remain in the
sample to age x+ 1. This situation is illustrated in Figure 6.3.
We note that ex and nx -Cx form separate binomial samples. A
member of Cx can die before age x+s, or can survive to that age and exit
observation; a member of nx -Cx can die before age x+ l, or survive to that
age and exit observation there. Let d; be the observed number of deaths out
of the Cx sample, and let d;' be the observed number of deaths out of the
nx -Cx sample, so that the total number of observed deaths is dx = d; +d;'.
Note also that ex = Cx -d; is the observed number of enders at age x+s.
The contribution to the likelihood of the ex sample is
(7.33)
and the contribution of the nx - Cx sample is
" - n-c-d"
Ln-c _
- (qx ) d . (1 qx )
·
(7.34)
Since the two groups are independent binomial samples, the overall likeli­
hood is the product of Le and L n -c· Thus
_ (sqx )d'. (!- sqx )c-d' . (qx )d" . (l- qx )n-c-d"·
L sqx
(7.35)
If this likelihood is evaluated under the uniform assumption, so that
s · qx, we find
(7.36)
Incomplete Data Samples: Ma.rimum Likelihood Procedures
156
Dropping the multiplicative constant sd ', and combining the two terms qd"
and qd ', we have
L
= qd . (1- s . qt-d ' . (1- qt-c-d ",
(7.37)
from which we obtain
s
c d")(-1 -) = 0.
din L = ef._ - (c-d' )(
q
1-s·q) (n- 1-q
dq
(7.38)
But c-d' = e, and n - c - d" = n - c - (d-d') = n - d - e, so we
see that (7.38) is the same as (7.31), so Estimator (7.32) will again result.
Thus we have found that the grouped Special Case C MLE under the uni­
form assumption is the same whether full data or partial data is available
with respect to the deaths.
If the likelihood given by (7.35) is evaluated under the exponential
assumption, with sPx = (Px f, we find
(7.39)
Using e = c - d', we see that the second factor in (7.39) can be written as
(px ye, and then combined with the last factor to produce
= p5r+n-c-d ". (I-pl". (1-yl',
l
(7.40)
from which we obtain
dlnL
dp
s
se+n-c-d" - d" - d' ·sp -l
1-p
1-ps
P
= 0.
(7.41)
The likelihood equation given by (7.41) will, in general, have to be solved by
numerical methods for the MLE Px (and hence qx = 1-px). A special case
exists ifs =
Equation (7.41) will then be quadratic in x = i1 2 and can be
solved for
The details are left as an exercise.
If the available information does not include the observed deaths d;
and d;' separately, nor show the subdivision of nx into Cx and n x-Cx , then a
more intricate partial data likelihood is created. We assume that the data
consist only of the facts that, out of n x persons at age x, there were ex
observed enders at x+s, d_, observed deaths, and nx-ex --d., survivors at age
x+l.
'A.
!;
157
Single-Decrement Environment, General Case
The expression
(7.42)
is not entirely correct for the likelihood, since we know that not all of the dx
deaths had the entire interval (x, x+ 1) in which to die (although we do not
know exactly how many did, and how many had only (x,x+s] in which to
die and be observed).
The unknown proportion of nx which are scheduled to exit at x+s
can be incorporated into the random structure of the model. Let ¢ represent
this unknown proportion; that is, ex = ¢ · nx. Then the probability of an
observed death is cp · sqx + ( 1-cp) · qx, and the first term of the likelihood
would be[¢· 5qx + (1-</>) · qxt, rather than merely (qx lWe would then proceed to find the MLE of cp, as well as the MLE of
qx. Note that cp appears only in the first term of L. Since In L is a decreasing
function of ¢, under both the uniform and the exponential distributions, it
follows that In L, and hence L, is maximized by cp = 0. Thus the same qx
which maximizes the correct likelihood will also maximize (7.42), so we
may work from (7.42) anyway!
When evaluated under the uniform assumption, (7.42) is easily seen
to be the same as (7.37), so Estimator (7.32) again results.
When evaluated under the exponential assumption, a nice simplifica­
tion occurs. We can write (7.42) as
(7.43)
and then combine the two terms in Px to obtain
L = pn-d-(1-s)e . (1-pl.
(7.44)
Then
dlnL _ n -d- (1-s)e
dp
p
d
1-p
0,
(7.45)
which solves linearly for
d
llx - (1-s)ex
(7.46)
We recognize (7.46) as the actuarial estimator for Special Case C, as des­
cribed in Section 6.4.
158
Incomplete Data Samples: Maximum Likelihood Procedure,
[(EXAMPLE 7.Slj Out of nx = 14,200 persons in a sample at age x, the sub­
set ex = 4400 are scheduled to end at age x+ { We observe d: = 360
deaths, prior to age x+i from the ex sample, and d;' = 3439 deaths, prior to
age x+ I from the remaining nx -Cx . Assuming that the force of mortality is
constant over (x, x+ I), find the MLE of qx .
!SOLUTION! We have c = 4400, n-c = 9800, d' = 360, d" = 3439, and
s = From (7.35), the likelihood is
t·
L = (1-p)3 439. (p)636t. (1-p t/4) 360 . ( pt/4)4040,
or, by combining the second and fourth terms,
L = (p)1311 . (1-p)3 439 . (1- pl/4)360.
The log-likelihood is
In L = 7371 · lnp + 3439 · ln(l-p) + 360 · ln(l-p 1 14),
and the derivative equation is
dlnL
dp
7371
p
3439
l-p
9op-3/4
1 - p l/4
0.
This leads to
7371(1-p)(l-p 1 14)- 3439p (l- p 1 14) - 90p 1 14 (l-p) = 0,
which can be written as
10,900x5 - 10,810x4 - 7461.x + 7371 = 0,
.
4
.
.
w I1ere x = p 11, 4 . An iterative
soIution y1e
. Ids x
/\ = P
/\ix / = . 90, so we h ave
/\
/\1/4 4
qx = 1- (px )
= .3439.
0
7.3.5 Special Case C with Random Censoring
As an alternative to recording, and using, exact or average values of s;, for
the ex persons with O < S; < 1, it is sometimes assumed that the values of S;
are randomly distributed over (x,x+ 1]. Then scheduled ending time is a
Single-Decrement Environment, General Case
159
random variable, S, and we let g(s) denote its PDF. In survival data analysis,
where termination of observation due to the ending of the observation period
is called censoring, this structure is referred to as a random censoring
mechanism.
To construct the likelihood for Special Case C with partial data, we
the
probability that a person in Cx will die before the scheduled ending
eed
n
age. If a precise scheduled ending age is not used, and a random distribution
of scheduled ending ages is assumed instead, then we must first calculate the
marginal probability of death before scheduled ending age. If we let cfx
denote this marginal probability, then
7Jx
= I - Px = l -
1
fo g(s) ·
sPx ds.
(7.47)
The integral in (7.47) shows that the marginal probability Px is a weighted
1
average value of sPx , where the sum of the weights is f0 g(s) ds = 1.
To evaluate (7.47), we must make a distribution assumption with
respect to mortality, and specify the distribution of Sas we! I. If we make the
linear assumption for mortality, and the uniform distribution for S (so that
g(s) = l ), then (7.47) evaluates to
(7.48)
The Special Case C likelihood is
(7.49)
under random censoring, as opposed to (7.35) for the likelihood when an
average (fixed) value of sis assumed. Of course if (7.48) is used for qx , then
(7.49) becomes the same as (7.35) evaluated under the uniform assumption
withs = and the resulting MLE is given by Estimator (7 .32) withs =
The evaluation of (7.47) assuming other mortality distributions, and
the resulting MLE's, is pursued in the exercises.
!,
!.
7.3. 6 Summary of Single-Decrement MLE 's
We have derived a general, full data, exponential distribution MLE, given by
Estimator (7.23), which, of course, is applicable to Special Cases A, B, and
Cas well.
i60
Incomplete Data Samples: Maximum Likelihood Procedures
Similarly, Estimator (7.27) is the general, full data, uniform distribu.
tion MLE, with Special Cases A, B, and C versions given by (7.7), (7.30)
and (7.32), respectively.
For partial data, we have developed results for Special Cases A and
C only, under each of the exponential and uniform distributions. The partial
data MLE's for Special Case B are derived analogously to those for Specia l
Case C, and are left to the exercises. The general case, partial data, MLE's
are considerably more complex, and are not pursued in this text.
7.4 DOUBLE-DECREMENT ENVIRONMENT
If both death and withdrawal are random events operating in the estimation
interval (x, x+ l ], then we seek to estimate q¥ = q�(d) within a double­
decrement environment. The basic mathematics and notation for doub le­
decrement theory was presented in Section 5.3, and employed in Section 6.3
in connection with moment estimation. Now we will explore maximum
likelihood estimation in a double-decrement environment.
7.4.1 General Form for Full Data
As before, let x+r; be the age at which person i enters (x, x+ I], 0 ::; r; < I,
and let nx be the total number of persons in the sample. Let x+t; be the age
at which person i leaves (x, x+ l ], 0 < f; ::; l, whether as an interval survivor
(t; = 1), as an observed ender (t; < l), or as a result of one of the random
events death or withdrawal.
Thus each person is under observation from age x+r; to age x+t;,
and the probability of this is 1,-rP�1r,· For interval survivors and enders, this
"total survival" (i.e., neither dying nor withdrawing) is the contribution to
the likelihood. For each death and withdrawal, the respective density func­
tions, given by (5.1 la) and (5.llb), are needed, so 1,-r,P�t, must be multi­
plied by the appropriate force. Thus the overall likelihood is
L
n
(r)
= TI 1,-r,Px+r,
ITw µx+t,
(w)
i=I
=
i(w)
· IT ,t'?
ITi=I. 1,-r;/Jx+r,
v
'1
f'X�I;
(7.SOa)
rr.
),\!
w
( )
µx+t,
(7.50b)
161
Double-Decrement Environment
Recall the indicator variable 15; defined by (7.17 ). We now introduce
a second indicator variable 1;, where
"f; = {
if person i withdraws in (x, x+ I J
1
1
l I.5 J )
f'J
O
otherwise
Using these two indicator variables, the likelihood can be written as
n
L = TI t,-r,Px+r, · 1,-r,Px+r, · ( µx+t,
1(d}
l(w)
(d}
i=I
µ�t
)/j' · (µ_,+t; ,
(w) )''
(7.50c)
is needed in the likelihood function only if person i is a death
since
(61 = I), and
is needed only if person i is a withdrawal(,;= 1).
Suppose our objective is to estimate q:<d)_ Since we have assumed
that death and withdrawal are independent, then the terms superscripted with
(w) are constant with respect to q�(d), and can be dropped from the likeli­
hood. Thus we have
µi1,,.
L =
n
i(d)
( (d)
TI t,--r,-Px+r, · µx+1,
- i=l
)6
1
(7.52)
Since the unadorned ,,- -rPx+r, and µx+t, in (7.20) are the same mortality func­
tions as are ,,- -r.P:�1,. and
in (7.52), it follows that (7.52) is exactly the
same as (7.20).
µ�t
7.4.2 Full Data, Exponential Distribution for Mortality
Since (7.52) is the same as (7.20), it follows that the maximization of (7.52)
under the exponential distribution will produce the MLE for µ<OJ, the con­
stant force over (x, x + I), given by (7.23 ).
7.4.3 Full Data, Uniform Distributionfor Mortality
Under the uniform distribution, (7.52), being the same as (7 .20), is maxi­
mized by the value of qx which satisfies
(7.53)
Incomplete Data Samples: Maximum Likelihood Procedurei
162
where the last summation is taken over all persons who do not die. Equation
(7.53), like its single-decrement counterpart (7.27), must, in general. be
solved by iteration.
A quadratic solution is found for (7.53) under Special Case A, with
r; = 0 for all i, I; = I for all enders, and t; = t, a constant, for all with­
drawals. (The derivation of this result is left as an exercise.) For all other
situations, we must solve a higher order polynomial equation to obtain
1
\ with the attendant possibility of multiple roots. Under the expo­
fJx =
nential assumption, on the other hand, all cases have a unique solution forµ
given by (7.23), and thus a unique solution for qr
q/
7.4.4 Partial Data, Special Case A
Estimation of q�(d) from partial data in the presence of random withdrawals
is more complex than with full data. We consider only Special Case A,
where r; = 0 for all i and s; = 1 for all i who do not die or withdraw.
Suppose our only information is that from a sample of nx persons at exact
age x, dx died and Wx withdrew in (x, x+ 1 ], so that nx -dx -Wx survived to
age x+ 1. Exact ages at death and withdrawal are not available.
The likelihood of this sample result is
_ [ (d)] d, [
L d
qx
·
(w)] w,
qx
(w)] n,-d,-w,
(
· [ I -qxd) -qx
,
(7.54)
w
where qi ) and q� ) are defined by (5.12a) and (5.12b), respectively. To find
the MLE's of qid) and qi"l, we first write
I ll L - dx . 1n qx(d) + Wx . 1 ll qx(w) + (nx -dx -Wx ) . I n (, I -q(xd) -·qx(w)) . (7.55)
Then we find
.4_ - nx -dx - Wx
I - q�d'J - q�vl
q�d)
0
(7.56a)
a lnL = W.r - nx - dx -- W.r
oqi"'i
I - qid'J - qiwl
qiw)
= 0,
(7.56b)
olnL
a qid>
and
which solve simultaneously for the expected results
Double-Decrement Environment
163
(7.57a)
and
(7 .57b)
Since our objective is to estimate q:(d} and q:(w), rather than merely
w
qidJ and qi ), we must once again adopt distribution assumptions for our ran­
do m events, death and withdrawal. This is pursued in the next two sub­
sections.
7.4.5 Partial Data (Special Case A), Exponential Distributions
From Equations (5.28a) and (5.28b), with qid) and qiw) replaced by their esti­
mators given by (7.57a) and (7.57b), we directly have
(7.58a)
and
q",x(w) = l - (n-d-w)w/(d+w).
n
(7.58b)
Note that MLE's (7.58a) and (7.58b) are the same as the moment estimators
(6.37a) and (6.37b).
7.4.6 Partial Data (Special Case A), Uniform Distributions
w
From Equations (5.20a) and (5.20b), with qicl) and ql ) replaced by their
estimators given by (7.57a) and (7.57b), we directly have
b-
J b -2nxdx
2
(7.59a)
(7.59b)
where c = nx -fir +fwx . Note that the MLE's (7.59a) and (7.59b) are the
same as the moment estimators (6.34a) and (6.34b).
Incomplete Data Samples: Maximum Likelihood Procedures
164
7.4. 7 Summary of Double-Decrement MLE 's
It should be recognized that it was the independence assumption for the
random events death and withdrawal which allowed us to reach the general
solutions for µ = i1> under the exponential distribution, given by (7.23),
under the uniform distribution, given by (7.53), for the full
and <Jx =
data case. [f the independence assumption is not valid, and a depende nt
model is assumed, then the estimation of qx is more complex. For a dis cus­
sion of this, the interested reader is referred to Robinson [63].
We chose to develop the partial data situation only for Special Case
A, since it is the only one of the special cases which has convenient solu.
.
.
.
c .,,,(d)
r/(w)
1or the ranttons 1or qx and qx . The more genera 1 partta 1 data s1tuat1on c
dom withdrawal model, which embraces all of our special cases, is given by
Broffitt [14].
d)
1ct
![EXAMPLE 7.6!j 100 persons enter the estimation interval (20, 2 l] at exact
age 20. There are two withdrawals, one at exact age 20.2 and the other at
exact age 20.7, and there is one death at 20.3. The remaining 97 persons
reach age 21. Estimate q;6d) under (a) full data, exponential distribution;
(b) full data, uniform distribution; (c) partial data, exponential distribution;
(d) partial data, uniform distribution.
[SQLUTION!
(a) From Estimator (7.23), µ = _20+_70�_30+97 = 9�_2, so we have
q;�> = l-e-11982 =.0101316.
(b) Looking at (7.53), we find r; = 0 for all persons, t; = I for 97 persons,
I;= .20 for one person, and t; = .70 for one person. Thus the likelihood
.
.
70
97
equation 1s qI - 1�_20
I · I I eads to a cub.1c
20q - 1�_70q - l-q = 0 , w11c1
equation in q. [f both witl-idrawals are placed at their average age of
2
20.45, then (7.53) becomes i - 1 �·.!� - t!.. = 0, which solves for
q
q
d
q;� ) given by (7.32), with ex replaced by Wx = 2, and withs== .45. This
2
.
'1,J(dJ 9.3
--=--iso - .0101106 .
gives q20 - 9 5- J<99.35)
90
(c) This is the same as the corresponding moment estimator, which was
1>
found to be
= .0 IO l O 17 in Example 6.7.
(d) This is also the same as the corresponding moment estimator, found in
N�
D
Example 6.6 to be q20 = .0101015.
q;6
Properties of Maximum Likeliltood Estimators
165
7.5 PROPERTIES OF MAXIMUM LIKELIHOOD ESTIMATORS
General properties of estimators are defined and summarized in Appendix A.
In this section we briefly observe the application of these properties to maxi­
mum likelihood estimators, with further details on the properties of MLE's
given in Appendix B.
7.5.1 Bias
The general conditions for asymptotic unbiasedness of an MLE are given in
Appendix B. In this section we will review the concept of bias by using the
familiar example of the MLE of the parameter a2 in a normal distribution. It
'£ (X;-X) , and that this is a
The unbiased estimator is fr� = n� 1 · '£ (Xi-X) .
is well known that the MLE is if�LE = � ·
biased estimator for 0' .
2
2
i=l
2
i=I
11� 1
• &�, so the MLE can be made unbiased by multi­
Then clearly &�LE =
plying it by the constant 11: . Furthermore,
1
(7.60)
Since Jim n-l = 1, this MLE is unbiased for infinitely large n, so we say it
n--+oc n
is asymptotically unbiased.
7.5.2 Variance
The asymptotic variance of an MLE of q,
q, is given by
(7.61)
if certain conditions pertaining to the likelihood function are satisfied. Equa­
tion (7.61) is derived in Appendix B.
.
To apply (7.61), the key step is taking the expectation of - a;;� L
We must identify the random variable(s) in - 8��� l, and know the expected
value(s) of such variable(s). Sometimes the expected value(s) will have to be
approximated, so (7.61) gives an approximate variance for this reason as
well as for its being used in conjunction with a finite n.
166
Incomplete Data Samples: Maximum Likelihood Procedures
jlEXAMPLE 7.711 Find the variance of the Special Case A estimator given
by (7.7).
[sOLUTION! Note first that Estimator (7.7) is a binomial proportion, so we
nd lnL = ?!..
q
know its variance is q(l- , d so
n )_ Now from (7.6) we have dq
q
-q'
2
·
1nL
d
d
n-d
d2lnL =
[
J
dq2
- 2 - (l-q)2, and we need E - � . Now d 1s actually the
q
realization of the binomial random variable D, and we should use D instead
of d when discussing expectations. We know that E [D] = nq and £[n-D]
= n(l-q). Substituting these expectations for D and n - D, we have
= nq n(l-q) = !1 + .....!!....__ = n(1-q) + nq =
E[-d 1nl]
+
2
q
l
2
dq
q2
(l-q)2
1-q
q( -q)
n
q(l-q)•
Finally, Var(q) is the reciprocal of this last expression, as already known.
Note that for this Special Case A MLE, (7.6 I) gives the correct variance for
finite n (i.e., not asymptotic only.)
0
[ixAMPLE 7.8!! Use (7.61) to approximate the variance of the Special
Case B MLE given by (7.30).
!SOLUTION! The left side of (7.28) gives us d��l for this case, so we be.
h
n-d
kr2
gm t ere. d 2 lnl = - d2 + l-r
and we see,k t he expecte
( q)2 - (l-q)2,
q
d q2
value of this expression. Note that only d (actually D) is a random variable.
However, since D is not binomial, then E[D] is not obvious. We see that
E[D] = bx · qx + kx · 1-rqx+r, so we need to make a distribution assumption
to approximate E[D]. Let us say E[D] � [bx + (1-r)kx] · qx , from the
hyperbolic assumption. Then
E[-d 2 1� L J
dq-
kr2 + n - [b + (1-·r)k]q
(1-rq)2
(1 - q)2
= n - rk
ta.2
n
_!El
-q- - (1-rq)2 + J - q + (l-q)2'
[b + (1-r)k]q
q2
since n = b + k, and Var(q) is approximated by the inverse of this expres­
sion. To numerically approximate the variance, we would substitute the
0
estimate q obtained from (7.30) for q in the variance expression.
Properties ofMaximum Likelihood Estimators
167
7.5.3 Consistency and Efficiency
MLE's are consistent estimators. Furthermore, within the class of asymptoti­
cally unbiased estimators, the variance of an MLE is as small or smaller than
the variance of any other estimator in the class, so the MLE is said to be an
efficient estimator.
7.6 THE PRODUCT-LIMIT ESTIMATOR
This estimator has been suggested by several writers, but is mainly attributed
to Kaplan and Meier [43), and is frequently referred to as the Kaplan-Meier
estimator. We have included it in our chapter on maximum likelihood esti­
mators since we will see that it makes use of a sequence of MLE's to
produce its estimate of qx , although the product-limit estimator itself is not
an MLE.
A basic characteristic of the product-limit estimator is that it does
not require a distribution assumption, such as the uniform or exponential.
7.6.1 Estimation of qx
The product-limit estimator can be used in any estimation situation, provided
full data are available. It treats enders and random withdrawals alike, and we
will use simply terminations to include both types. The presence of entrants
at any age is also handled by this estimation approach.
The product-limit estimator is most easily explained by an example.
Suppose a sample of nx = 20 lives at age x is augmented by 2 additional
entrants and decremented by 3 terminations and 5 deaths prior to age x+ 1.
Figure 7.1 illustrates this data, using arrows to locate the entrants and
terminations, and X's to locate the deaths.
J. 20
I
x
,,¥
i
L
k
)f
1
1
,¥=
1
.,¥
I
x+l
!FIGURE 7.11
We have not bothered to indicate the ages within (x,x+l] at which the
events of death, entry and termination occur, since this information is not
used in computing cix - It is only necessary to indicate the events in order of
occurrence, but not the precise ages at which they occur.
Incomplete Data Samples: Maximum Likelihood Procedures
168
We now break our basic estimation interval into six subintervals by
partitioning (x, x+ I] at each point where an entry or termination occurs, as
shown in Figure 7.2.
120
I
x
v l
7i
l
x x
l
l
x
x+1
!FIGURE 7.2!
We recognize that each of the six subintervals represents a Special Case A
situation, since there are no events within the subinterval except (possibly)
deaths. If q; is the conditional probability of death in the i1h subinterval,
given alive at the beginning of that subinterval, then the MLE of q; is the
simple binomial proportion
d;
n; '
(7.62).
where d; is the number of observed deaths in Subinterval i, and n; is the bi­
nomial sample size for that subinterval. Clearly if d; = 0, as is true for i = 2
and i = 4, then cf; = 0 as well.
The product-limit estimator of qx is defined as
(7.63 )
where mis the number of subintervals in (x,x+1]. (m = 6 in our example.)
To complete the example, we first note that the subinterval sample
sizes are n1 =--" 20, n2 = 20, 113 = 19, n4 = 16, ns = 17, n6 = 15. Thus we
(16)(14)
16) ( 16) ( 14) _
_ J - ( 20
17) ( T6
I,
/\ 19) (20)
20 ('T9
TI Ts - I - (20)(15) _
1ave q,
- .2::i333.
The occurrence of a death at the same age as an entrant or termina­
tion raises the question of the subinterval to which such death belongs. The
usual rule is to consider a death which coincides with an entrant or termin­
ation as belonging to the subinterval preceding such event. That is, a death
at a subinterval boundary is deemed to belong to the subinterval closing at
the boundary, rather than the one opening there. This is consistent with our
general notation (x, x+1].
The Product-Limit Estimator
169
As an alternative to partitioning (x,x+ 1] at each entry and termina­
tion point, we could partition at each death point instead. In that case our
example would result in the six subintervals illustrated in Figure 7.3.
120:
xf
. l
}f:
l
l
l
x+l
!FIGURE 7.3!
We call r1 the risk set for the fh subinterval, and define it to be the number
of persons in the sample immediately preceding the death which marks the
end of the /h subinterval. The estimate of the probability of death in the fh
subinterval is then
/\
qj
1
= 0'
(7.64)
except for the last subinterval in (x,x+ I]. Since ther� is no death in that
subinterval (Subinterval 6 in our example), then the q1 for that subinterval is
zero. Finally,
7
t = 1 - TI (1-q) = 1 - TI (r-1)
gives the estimate of qx .
m
m
J=I
j=I
J
(7.65)
!!EXAMPLE 7.9!1 Show that the same estimate of qx results for partitioning
at death points as for partitioning at entry and withdrawal points.
!SOLUTION! From Figure 7.3, we find the risk sets to be r 1 = 20, r2 = 19,
r3 = 18, r4 = 17, r5 = 15. Then from (7.65), we find the product-limit esti/\
6)(14)
6 14) = 1- (1
18) ( 17) ( 17
0
19 ) ( 19
mate qx = 1- ( 20
(lO)(lS)
= .25333.
18 1 ) ( 15
In the partitioning-at-death approach, if a death and entrant, or death
and termination, occur at the same age, we need rules to determine the risk
set associated with that death. In order to interpret the risk set as existing
"immediately preceding" the death, we treat the death as preceding the
entrant or termination. Thus a simultaneous entrant is not included in the
risk set, whereas a simultaneous termination is included (i.e., termination has
not yet taken place).
170
tied to
Incomplete Data Samples: Maximum Likelihood Procedures
If there is more than one death at the same age, then (7.64) is modi. 6)
(76
where cl; is the number of simultaneous deaths marking the end of the /h
subinterval.
Note that if there exists any subinterval for which a study sample
does not exist, then the qj for the subinterval does not exist, and so f:ix does
not exist as well. This occurs if all members in the sample terminate before
agex+I.
7.6.2 Properties of tlie Estimator
The product-limit estimator is unbiased and consistent.
For the variance of the estimator, let us consider
/\
m
Px = I1P
/\ ;,
i=l
(7.67)
P;
which comes directly from (7.63). Recall that each
is a binomial pro­
portion,conditional on its sample size 11;. Then allp/s are conditionally mu­
tually independent, and, being binomial proportions, they are unbiased.
Therefore, the conditions of Section 6.5 hold here, so by steps analogous to
those leading to Equation (6.45), we find
(7.68)
Sinceqx = 1- px ,itfollows that(7.68)is Var(qx l{n;})as we11.
By the same steps as those leading from (6.45) to (6. 46), we can
obtain the Greenwood approximation
2
Var(qx l{n;}) = Var(px l{n;}) � (Px) ·
L(p%J·
m
(7.69)
i=l
In either case, a numerical estimate of the variance is obtained by
substituting the estimates
and x just obtained from the sample, along
with the n ;'s from the sample.
For a very large sample with many entrants and terminations, such as
found in insurance company studies, the product-limit estimator is not
P;, q;
P
The Product-Limit Estimator
171
generally used, due to the large number of subintervals that would result.
For clinical studies, however, it is commonly used, especially for those
situations in which there are no entrants after time t = 0. We will now
explore this use of the estimator.
7. 6.3 Estimation ofS(t)
Let us assume a sample of n persons all observed from time t = 0 on the
follow-up scale. Ifthe study is halted before all have died, then we will have
terminations from the sample at various values oft.
In clinical studies it is customary to use the approach of partitioning
at each death point. Then ifri is the risk set immediately preceding the lh
death point (which, in general, has c4 deaths, with c4 most frequently being
one), we have, as before,
0_
r· .
(7.66)
J
Note that qi estimates the probability of death over the interval which ends at
the fh death point, given alive at the beginning of that interval; it is not the
probability ofdeath over an interval which begins at t = j. Then
S(tm ) =
(I-qj) = IT (ri;.0)
IT
J
j =I
j=I
(7.70)
estimates the probability ofsurvival from I= 0 tot= Im , the time of the m'h
death point. For all I such that Im � t < lm+I, the estimate of S(t) is the
A
same as S(tm ), since there are no deaths in the sample between tm and t.
Thus
S(t) =
"
for Im� t < lm+I, m = 1, 2, . ...
IT (r·-d·)
�
J
m
j =I
(7.71)
As a special case, S(t) = I for t < t 1•
A
Analogous to (7 .69), the variance ofS(t) is approximated by
Var [s(t)l{rj }] � [S(t)]2 .
m
L(J
t.),
J
.= I
'J J
(7.72)
Incomplete Data Samples: Maximum Likelihood Procedures
172
for Im :S t < Im+ 1/\
If ( 7. 72) is then numerically estimated by substituting qj
/\
for qJ, PJ for PJ, and S(t) for S(t), we obtain
.
/\
smce q1 = <!.J_
r,
Finally, if there are no terminations (i.e., complete data), then, for all
j, r1+ 1 = rJ -�- It is then easy to see that (7.70) becomes
= rm+I
m
rj+l
S(tm ) = rr ( r· )
J
I
r1 '
}=
(7.74)
where r 1 = n, the original sample, and rm+ t is the number of survivors just
preceding the (m+ I )51 death point, or just after the m1h death point. Thus
S(tm ) is estimated by the proportion surviving to that time, and we have
returned to the approach of Section 4.3.
7.6.4 The Nelson-Aalen Estimator
Recall Equation (2.13) which states that·S(t) = e-A(rl, where A(t) is the cum­
ulative hazard function. It would be reasonable to estimate S(t) by first esti­
mating A(t) and then defining
(7.75)
S(t)
From (7.75) we have the relationship
/\
A(t) =
/\
- In S(t).
(7.76)
Consider again Equation (7.70), the product-limit estimator of S(t),
for Im :S t < lm+l· Substituting (7.70) into (7.76) we have
Ac,) - -
1{�, ( �)]
m
-Lin ( 1;=I
d), t,,,�t<lm+I·
0
(7.77)
"
,j
17
The Product-Limit Estimator
cf1)
cf1)
2
Recall that - In ( I - r1 = �
r1 + -2 ( r1 + · · · . By ignoring quadratic and
higher power terms, we obtain the approximation
1
(
d·) � ..1..
d·
- In 1- ..1..
rj
rj'
(7.78)
thereby approximating the cumulative hazard by
m
d·
L
i
, lm :SI< lm+I,
j=I J
/\
A(t) =
(7.79)
and, in tum, approximating the survival function by
/\
( � di·) ,
S(t) = exp - �
j=I J
Im :St< lm+I,
(7.80)
which is called the Ne/son-Aalen estimator (see Nelson [57)).
If there is only one death at any death point (i.e., 0 = I for all j),
and if there are no terminations, then the Nelson-Aalen estimator is particu­
larly easy to calculate. In that case, rj = n-j+ I, so
/\
A(t) =
L
l = k + n � l + · · · + n _ ! + l , lm :S t < lm+I·
j=I
m
(7.81)
II EXAMPLE 7 .toll In a complete data study, the Nelson-Aalen estimate of
A(t) immediately following the second death is 11/30. Find the estimate of
A(t) immediately following the fourth death.
!SOLUTION! After the second death A.(t) = * + n� 1 , where n is the
�6,
original size of the sample. We are given that A(t2) =
which implies
2
�+ n� l = � ::=/) =
or l ln -7ln+30 = 0, which solves for n = 6.
�6,
(The other root, n = ?, , is extraneous since n must be an integer.) Thus
"
0
A(t4) = 6I + 5I + 4I + 3I = .95.
0
The relationship between the product-limit and the Nelson-Aalen es­
timators is illustrated by the following example.
174
Incomplete Data Samples: Maximum Likelihood Procedures
!!EXAMPLE 7.111! In a certain complete data stu<ly with original samp le of
size I 0, the product-limit estimate of S(l 2) is found to be S( 12) = . 70. Find
the Nelson-Aalen estimate of S( 12).
!SOLUTION! Since we have complete data, there must be 7 survivors at
t = 12 (since the product-limit estimator is a simple binomial proportion in
this case). Thus there have been 3 deaths by t = 12, so the Nelson-Aalen
/\
/\
-242/720
·
·ts A (l2)
est11nate
=. 71 454.
= 10I + 9I + 8I = 242
720, and so S( l2) = e
(Note that the Nelson-Aalen estimate exceeds the product-limit estimate; see
Exercise 7-30.)
0
7.7SUMMARY
In this chapter we have developed MLE's for 9x (or for the constantµ under
the exponential assumption), for various combinations of decrement environ­
ment (single or double), type of data (full or partial), and distribution as­
sumption (exponential or uniform). Table 7.2 summarizes the reference
numbers of most of the estimators developed in this chapter.
Single-Decrement
General Case
Special Case A
Special Case B
Special Case C
Double-Decrement
General Case
Special Case A
Special Case B
Special Case C
TABLE7.2
Full Data Full Data
Exponential Uniform
Partial Data
Exponential
Partial Data
Uniform
-
-
(7.23)
(7.15)
(7.23)
(7.23)
(7.27)
(7.7)
(7.30)
(7.32)
(7.41) or (7.46)
(7.23)
(7.23)
(7.23)
(7.23)
(7.53)
-
-
(7.58)
(7.59)
**
(7.53)
(7.53)
Notes: 1 No distribution assumption is actually made.
2 (7.46) is the same as the actuarial estimator.
* See Exercise 7-1 L
** See Exercise 7-18.
(7.7)1
(7.7)1
*
*
-
2
(7.32)
-
-
175
Exercises
In addition to these MLE's, we developed product-limit estimators
given,
equivalently, by (7.63) and (7.65), and for S(t) given by (7.71).
q
for x
The Nelson-Aalen estimator for S(t), given by (7.80), is an alternative to
(7.71).
7.8 EXERCISES
7.1 Introduction; 7 .2 Single-Decrement Environment, Special Case A
7-1. Find an equation which is satisfied by the MLE of qx for Special Case
A, full data, under the hyperbolic assumption.
7-2. Solve the equation of Exercise 7-1 for qx if all deaths occur just before
age x+1 (i.e., ifs; = 1 for all deaths).
7-3. Intuitively, if there are no deaths in our sample we might expect to
find qx = 0. Similarly, if the entire sample of nx persons dies, we
might expect to find
= 1. It is obvious that these results are
obtained for the partial data and the full data, linear distribution,
situations. Are they obtained for the full data, exponential distribution,
situation?
t
7-4. (a) Express the likelihood given by (7.11) in terms of qx under the ex­
ponential assumption.
(b) Express the log-likelihood for the likelihood in part (a).
(c) Solve this log-likelihood for the MLE of qx.
7-5. Over the estimation interval (x,x+1] it is given that sPx = 1 - 3{ (qx)2
for O � s �
and that sPx = 1 - s(qx)2 for
s < 1. If nx = 300
and there are two observed deaths, one at x+.40 and one at x+. 75, find
the MLE of qx.
i,
i ::::;
7.3 Single-Decrement Environment, General Case
7-6. Show that the Special Case C, partial data, uniform distribution likeli­
hood, given by (7.36), leads to Estimator (7.32).
176
Incomplete Data Samples: Maximum Likelihood Procedures
7-7. Table 7.3 gives the data for a sample of five lives over the estimation
interval (x, x+1].
TABLE 7.3
i
r;
f;
b;
1 0.0 l.0 0
2 0.0 .6 l
3 0.0
.7 1
.5
4
.9 1
5
.5 l .0 0
Find the MLE of qx under each of (a) the exponential distribution, and
(b) the uniform distribution.
7-8. A sample of 105 individuals were alive at age x. Five died before
age x+1, and 10 others were enders at age x+
If the force of mor­
tality is constant over (x, x+ 1 ), find (a) the actuarial estimator, and (b)
the maximum likelihood estimator of qx.
i.
7-9. Refer to the Schwartz-Lazar moment estimator of Section 6.2.6.
(a) Of the Cx group, what is the probability of being
(i) an observed death?
(ii) an observed ender?
(iii)a survivor to agex+l?
(b) Write the likelihood, Le , for the Cx group.
(c) Of the (nx -Cx ) group, what is the probability of being
(i) an observed death?
(ii) an observed ender?
(iii)a survivor to agex+1?
(d) Write the likelihood, Ln-c, for the (nx �Cx ) group.
(e) Write the overall likelihood, and show that the MLE of qx is the
same as the Schwartz-Lazar estimator given by (6.28). (Note the
similarity of (6.28) and (7.32).)
7-10. Solve Equation (7.41) forpx using s =
!-
7-11. Derive the partial data, Special Case B MLE for qx , assuming (a) the
uniform distribution, and (b) the exponential distribution with r; =
for all i.
!
Exercises
177
7-12. Evaluate (7.47) forqx by assuming a hyperbolic fx+s for mortality, and
the uniform distribution for S.
7-13. Evaluate (7.47) for cfx by assuming an exponential fx+s for mortality,
and the uniform distribution for S.
7-14. The expression for Px found in Exercise 7-13 is commonly approxi­
mated bypx :=:::: // . Derive this approximation.
2
7-15. The Special Case C likelihood under random censoring is given by
(7.49). If we assume that scheduled ending ages are uniformly distrib­
uted, but mortality follows the exponential distribution, then we have
cfx ::::::: 1 - ( l-qx ) 1 12 (see Exercise 7-14). Derive the MLE of qx under
these assumptions. (This is called Drolette's Estimator B [23].)
7.4 Double-Decrement Environment
.n(r) , show that IX
q(r) = Iqx1 <d) + Iqx'(w)_ I qx,(d) · t qx,(w), and
q(r) = 1- lrx
7-16 • If IX
justify this relationship by general reasoning.
7-17. Suppose there are nx persons in a study sample. Let ri be defined by
(7.S1), and let 8-; be defined by (7.17). Show that the general form for
the full data likelihood, given by (7.50b), can also be written as
L =
(d)]
E� ,.P'x
n [
1-6;
[
d
,(d)
. - dt tPx lt=l;
]li' . tPx
[
,(w)] 1-"Y;
[
d
i(w)
. - dt tPx lt=I;
]'Y;.
7-18. Solve equation (7.53) under Special Case A with all withdrawals oc­
curring at f; = t, a constant.
7.5 Properties of Maximum Likelihood Estimators
7-19. Approximate Var(qx) for the Special Case B Estimator (7.30) by using
the linear assumption to approximate E [D], rather than the hyperbolic
assumption as was done in Example 7.8.
7-20. Find the asymptotic variance ofµ given by (7.15).
178
Incomplete Data Samples: Maximum Likelihood Procedures
7-21. Find the asymptotic variance of the estimator derived in Exercise 7-15.
[Note that, under the assumptions of the model, E[D'] = c(l-p 1 i2 )
and E[D"] = (n-c)(l-p).]
7-22. Find the asymptotic variance of the Schwartz-Lazar estimator derived
as an MLE in Exercise 7-9.
7-23. Consider the Special Case A, partial data, situation of Section 7.4.4.
Note that both D and Ware random variables.
(a) What is E [DJ?
(b) What is E[W]?
(c) Under the exponential distribution assumption, let a = µ<d) and
. d cJ2 lnL 82 lnL and 82 InL
/3 = µ(w) · F111
80? ' 8(32 '
80.8(3 ·
8
(d) (i) Find E[
��/], and let it be the element/11 of matrix/
-
�1/], and let it be the elementfi2 of matrix/
(iii)Find E[ - i��J], and let it be the elements /12 and Ji 1 of
(ii) Find E [ - 8
matrix/
(e) The inverse of matrix f, g = 1- 1 , gives Var(�) as element g11,
/\
/\ /\
Var(/3) as element g22, and Cov(a,/3) as elements g12 and g21.
/\
/\
""
Find Var(a), Var({3) and Cov(a,/3).
. ·
7.6 The Product-Limit Estimator
7-24. Show that in a Special Case A, full data, situation, the product-limit
estimator will be qx = �, the same as (7. 7) and ( 6.7).
nx
7-25. One hundred annuitants are observed from attainment of exact age 75
for part or all of the following year of age. There are four deaths, one
at the end of each quarter year of age. At the end of the observation
period, there were ten annuitants aged 75 j, and the other survivors had
all attained age 76 or greater.
(a) Find the product-limit estimate of q7s.
(b) Estimate the variance of q75 by using q; in place of each q;.
179
Exercises
7-26. In a clinical study with n lives at t = 0, there are no intermediate
entrants. There was one death at time t7, two deaths at time t8 , and one
death at time t9 . The product-limit estimate of S(t) is found to be
/\.
/\
/\
S(h) =. 75, S(ts) = .60, and S(t9) = .50. Detennine the number of
terminations from the study between times t8 and t9.
7-27. Ten nuclear power plant workers were accidentally exposed to a signi­
ficant dosage of radiation. A death is observed at each of times 2 and
4, and x withdrawals are observed at time 3. Using the product-limit
/\.
estimator of S(t), it is found that S(S) = .75. Determine x.
r-28. Consider a newborn litter of five mice. As a result of the poor health
of the mother, they die at times 2, 3, 6, 9, 12. Estimate both S(lO) and
A(IO) by (a) the product-limit method, and (b) the Nelson-Aalen
method.
7-29. In a complete data study, with only one death at each death point, A(t)
/\.
is estimated by the Nelson-Aalen method. We are given A(tk) = .3101
and A(tk+i) = .3726. Find A(tk+2).
/\
/\
7-30. If PLS(t) and NAS(t) represent the product-limit and Nelson-Aalen esti­
mators of S(t), respectively, show that PLs(t) < NAS(t), for all t 2: t 1 .
CHAPTERS
ESTIMATION OF PARAMETRIC SURVIVAL MODELS
8.1 INTRODUCTION
We now turn our attention to the estimation of survival models in para­
metri c form. Fewer quantities are to be estimated here than was the case for
our tabular models, as described in Chapters 6 and 7.
We will consider two families of parametric models in this chapter.
Most of the chapter will deal with estimating the parameters of our familiar
univariate model S(x) or S(t). In tum, we will explore this estimation problem
under both complete data and incomplete data samples. The second kind of
parametric model to be considered is the one which involves concomitant
11ariables. This will include the select model S(t;x), which arises in an
actuarial setting, and the more general model S(t;z), which arises in a clinical
setting.
Various approaches to parameter estimation will be considered, with
emphasis given to least-squares and maximum likelihood methods. Proper­
ties of the estimators will be examined, along with some hypothesis testing
of the chosen model and its parameters.
8.2 UNIVARIATE MODELS, COMPLETE DATA
As we saw in Chapters 4 through 7, estimation is generally more straight­
forward with complete data samples. As in Chapter 4, we will distinguish
between samples in which exact times of death are known, and those in
which times of death have been grouped. We will use S(t) for the underlying
survival model to be estimated, rather than S(.x), to suggest that complete
data samples generally exist in a clinical setting.
8.2.1 Exact Times of Death
Suppose a sample of n lives, .all existing at t = 0, produces observed times of
death t 1, t2, ···,In, assumed to be independent. We wish to use this data to
estimate S(t) as a parametric survival model. In other words, we adopt a
particular mathematical function of the variable t, depending on one or more
parameters as well, and then use the sample data to estimate these unknown
parameters, using a particular estimation procedure.
Estimation of Parametric Survival Models
182
The simplest parametric models to use are those which depend on
only one parameter, such as the exponential distribution with S(t) = e->.,.
How can we estimate >. from our sample data?
Let us calculate
l.�f
n L''
t
(8.l)
i=l
which is the mean time of death of the sample. We know from (2.29) that
the mean of Tis E[T] = if T has an exponential distribution, so we can
estimate >. by equating these two means. Thus
f
� = } =
LI;.
i=l
(8.2)
This estimation procedure is called the method of moments, so Estimator
(8.2) is the moment estimator of>..
I EXAMPLE 8.l!I Show that Estimator (8.2) is the maximum likelihood esti­
mator of>. as well.
!SOLUTION! The likelihood for the i1h death is the PDF for death at time t;,
f(t;) = >.e->.1;. The overall likelihood is given by L = [I >.e->.\ and the loglikelihood is lnL = n ·In>. - ). ·
leads to the same result as (8.2).
f,
i=l
t;. Then d��L =
i=t
X - i=l
"E,t; = 0, which
D
Since Estimator (8.2) is the MLE of,\, we know it is asymptotically
unbiased, with asymptotic variance given by
(8.3)
as stated in Section 7 .5 .2. The reader ts asked to find the variance of
Estimator (8.2) in Exercise 8-1.
Univariate Models, Complete Data
183
As an alternative to the method ofmoments, which equated the mean of
the distribution (
to the mean of the sample ( t), we might instead equate
±)
the distribution and sample medians. Let 7 represent the sample median,
and let t 1 < t2 < · · · be the observed values of t in increasing numerical
order. Ifn is odd, for example n = 25, then 7 = t 13 is the sample median; if
n is even, for example n = 24, then 7 =
12 + t13) is usually taken as the
median. On the other hand, the median of the distribution is that value oft,
or e->.1111 = under the exponential distrit 1 ;2 say, for which S(t 1;2) =
1U
1,
bution. Solving for t 1 ;2 we find that t 1 ;2 =
results from equating t 1 ;2 and 7. Thus
!1;2
producing
=
In l
.:::...2..
/\
.\
1
-¥,
and our estimate of,\
= 7,
.\
(8.4)
In l
.:::...2..
(8.5)
7·
!!EXAMPLE 8.211 A sample of 10 laboratory mice produces the times of
death (in days) 3, 4, 5, 7, I, 8, 10, 10, 10, 12. Assuming the operative survi­
val model to be exponential, estimate ,\ by both the method ofmoments and
the method ofmedians.
!SOLUTION! The sample mean is t = 7.6 and the sample median is 7 = 7.5.
/\
Then ,\
= 1I = .13157 by the method ofmoments, and ,\ = -f2 = . 09242
by the method ofmedians.
0
/\
- In !
For a parametric model with two parameters, the method of mo­
ments equates the sample and distribution means, and the sample and distri­
bution second (central) moments. For our sample of times of death ti,
i = I , 2,- · ·, n, the sample second (central) moment, s2 say, is given by
1
s2 = -n- l .
L (t--t)-2
I
'
(8.6)
i=I
where t is the sample mean. Then ifµ and a2 are the distribution mean and
variance, respectively, in terms ofthe two unknown parameters of the distri­
bution, we would find estimates of those parameters by solving simultane­
ously the equations µ = t and a2 = s2 .
184
Estimation of Parametric Survival Modelr
This method of parameter estimation is facilitated ifwe have simple
expressions for µ and a2 in terms ofthe distribution parameters. Ifthese do
not exist, then the moment equations may require a numerical solution, as
would be the case for the popular actuarial parametric models of Gompertz
and Makeham. On the other hand, in some cases the moment equations
might not have a solution at all, so this method ofparameter estimation can­
not be used.
\!EXAMPLE 8.3!j The two-parameter gamma distribution is defined by its
PDF as j(t) = {Joi(a) l1 - 1 e-t/(J, t > 0, a > 0, f3 > 0, and its mean and var­
iance are µ = f3a and a 2 = f32 a. Using the sample data of Example 8.2,
estimate o: and /3 by the method ofmoments.
!SOLUTION! We have t = 7.6 from Example 8.2, and we find the sample
IO
I · I)t;-7.6)
2
= 7.84. Then our moment equasecond moment to be s2 = 10
i=I
tions are f3a = 7.6 and f32 o: = 7.84. Next we have 7.84 = f3(f3o:) = {3(7.6),
"
7.84
so f3 = TJ;
= 1.03158, and o:" = 7.6
� = 7.36734.
D
For two parameters we might also use the method of percentiles,
which is an extension of the method of medians (the median is the 501h
percentile). If tp is a value oft such that S(tp ) = l -p, then Ip is the I OOpth
percentile ofthe distribution. Ifp = then tp is the median, t 1 ;2• Note well
that ifS(tp) = .95, for example, Ip is the 5th percentile, not the 951h . We then
observe (or approximate) the corresponding percentiles in the sample, equate
them to the distribution percentiles (which are in terms of the unknown
parameters), and solve for the parameters. An example will make the proce­
dure clear.
f,
\!EXAMPLE 8.4lj Estimate the Gompertz parameters Band c (see Equation
(2.32)) from the sample data of Example 8.2 by the method of percentiles,
using the (approximate) 251h and 65 1h percentiles.
[[.OLUTION! From the data, let us approximate the desired sample percen­
tiles as 4.5 and 9. (The reader should verify the reasonableness of these
values.) For the distribution, S(t_25) = exp [ ,:c (1-c' 25)] = . 75, so
Univariate Models, Complete Data
t.2s =
In [1 _
·In
�
c]
Inc
Similarly,
1·6:i_ =
In( .7
185
In [1 _
In( .3
·In
�
c]
·
Inc
When we equate ( 2 5 = 4.5 and t.65 = 9, the resulting equations produce
/\
B = .03777 and c = l.24173. The details of this solution are left as an
A
D
exercise.
The method of maximum likelihood may also be used for models
with two parameters. Here we would take the partial derivative ofIn L with
respect to each parameter, set each derivative to zero, and solve the resulting
pair of equations simultaneously. Frequently these equations are not con­
venient ones in the unknown parameters, and a numerical solution must be
obtained.
HEXAMPLE 8.sjj Estimate the Weibull parameters k and n from the sample
data of Example 8.2 by the method of maximum likelihood.
10
10
�·
bl
!SOLUTION! Recall that L = ITf(t;), so that lnL = L lnf(t;). The PDF is
f(t) = S(t) · ,\(t), so, from (2.35) and (2.36) we find the PDF of the Weibull
Then the- log-likelihood is
distribution to be f(t) = k · f' · exp [ IO
10
i=I
i=I
\�;1] .
In L = 10 · Ink+ n · L In I;- n +k 1· Lt'/+ 1. Differentiating, we obtain
lnL = "'Int· aQn
�
10
r=I
I
10
10
k
r1I +1 - "'11I +1
(n+I)·"'
I
� (Int)·
�
(n+})2
r=t
r=I
as well as
olnL
ffk
[
IO
I . "'..n+I _ o
lQ __
_
n+ I �';
k
i=l
l
= 0,
186
Estimation of Parametric Survival Models
I). This is substituted
From this second equation we then find k = 101�(n+
t;+l
�
i=I
into the first equation which is then solved numerically for�- Finally, this�
is used to find k.
D
We next consider a technique of parameter estimation that does not
directly use the observed times of death. Suppose we have produced, from
the exact times of death f;, i = l,2,- · ·,n, the estimated survival distribution
S0 (t) given by Equation (4.1), and illustrated in Figure 4.2 for a sample with
n = 5. Note that this S0 (t) is "parametric" in the sense that it is defined for
all t � 0, but not in the sense that we are using the term in this chapter.
Since S0 (t) is a representation of the survival pattern observed for the sample
to which it relates, it is frequently called the observed survival function,
which explains the superscript "o." We will now use S(t) for our parametric
model with estimated parameters; the distinction between S(t) and S0 (t)
should be clearly understood.
We assume a parametric model for S(t), with unknown parameters,
of course. We could then consider the sum of squares
SS =
L [S(t;) - S (t;)] .
n
0
2
i=I
(8.7a)
The only unknowns in (8.7a) are the parameters in S(t;), so we could differ­
entiate SS with respect to each parameter, set each derivative to zero, and
solve for the least-squares estimates of the parameters.
To illustrate, let us fit a uniform distribution (with unknown w) to
the data of Example 8.2. S0 (t) for this data is shown in Figure 8.1, with the
adopted uniform S(t) superimposed. For this data, although there are IO
deaths, there are only 7 distinct times of death, due to the multiple deaths at
t = 7 and t = 10.
Note that Equation (8.7a) defines the deviations, whose squares are
to be minimized, to be those between S(t) and S0 (t) at the bottom left corners
of S0 (t). (E.g., t 1 = 3 in Figure 8.1, and S0 (3) = .90; similarly, h = 12 and
S0 (12) = 0.) This procedure exerts a downward bias on(':;_ Conversely,
SS =
L [S(t;) - S (t;_i)]
n
i=l
0
2
(8.7b)
Univariate Models, Complete Data
187
fits S(t) to S 0{t) at the top right corners of S0(t}, thereby exerting an upward
bias on [;;_ The best procedure is to fit S(t) to S0 (t) at the midpoints of the
vertical line segments in Figure 8.1. Thus
SS =
L { S(t;)- ![S (t;) + S (t;-1))}
n
0
0
2
(8.7c)
i=I
will be minimized to find the least-squares estimates.
1.00
.90
S"(I)
�
.80
.70
.60
.50
.40
.30
.20
.10
3
0
4
7
5
8
10
12
w
!FIGURE 8.11
have
Applying (8.7c) to find D from the data shown in Figure 8.1, we
(8.8)
Differentiating, we find
dSS
dw
or
7
°L'i -
i=I
di· I:tt- I:>;· !cs
7
7
i=I
i=d
0( )
0
t; +s (t;-1)1 - o,
Estimation of Parametric Survival Models
188
or
7
'f:,tf
=
/\
w
7
7
i=l
i=l
i=l
I:, l; - L, I; ·
Hs (t;) + S (t;-1 )]
0
0
.
(8.9)
Table 8.1 gives the values needed to evaluate (8.9) for[;;_
TABLE8.l
i
I
2
3
4
5
6
7
S (t;)
.90
3
4
.80
5
.70
.50
7
.40
8
.10
10
12 0.00
49
0
I;
I�'
9
16
25
49
64
100
144
407
I;· f[S0 (t;) +S0 (1;-i))
2.85
3.40
3.75
4.20
3.60
2.50
.60
20.90
Then D = 49 �0J0_9 = 14.48399.
Least-squares estimation of parameters is facilitated if the parame­
ters are linear within SS, so that the derivative equations are linear. Since
this is frequently not the case, a transformation, usually logarithmic, might
be used.
!!EXAMPLE 8.61! Fit the data of Example 8.2 to an exponential distribu­
tion, estimating ). by the method of least squares.
!SOLUTION! We have
7
2
SS= I:{e-Al;_![S0 (ti)+S0 (t;_i)]},
i=l
(8.10)
so �s = 0 will be an exponential equation in >.. To avoid this we will fit
lnS(ti) = -- Al; to tn{
Hs (t;) + S (tH)]}. We now have
0
0
189
Univariate Models, Complete Data
(8.lOa)
which leads to
dlf = I::(7
2·
i=I
or
,\t; - In{ ![S0(ti) + S 0(t;_i)]}) (- I;) = 0,
7
7
i=I
i=I
,\·LI;+ LI;· ln{![S°(t;)+S (t;_ 1 )J} = 0,
0
(8.11)
(8.lla)
which produces the estimator
,\
t
t=I
I; · In{ 1(S°(I;) + S0{1;-1)]}
7
I:tl
i=I
(8.12)
"
Table 8.2 gives the values needed to evaluate (8.12) for,\.
;
I
2
3
4
5
6
7
Then� = 62
t; S (t;)
.90
3
.80
4
.70
5
.50
7
.40
8
.10
IO
12 0.00
49
0
TABLE8.2
tf t; · In{ l(S0 (t;) + S0 (t;-1))}
- .15387
9
- .65007
16
- 1.43841
25
49
- 3.57578
64
- 6.38806
100
- 13.86294
144
- 35.94879
407
- 62.01792
·JM92 = .15238.
0
For some survival models, such as Gompertz and Makeham, a trans­
formation to make S(t) linear in its parameters does not exist. The method of
least squares could still be used, of course, with the derivative equations
being solved numerically, but some of the advantages of this method of
estimation would thereby be lost.
Estimation of Parametric Survival Models
190
Note that unweighted least squares were used in our discussion.
If values of S0 (t;) are felt to be more reliable for some l; than for others,
2
we could use greater weights on the { S(t;) - !(S0 (t;) + S0 (t;_1)]} squared
deviations for such values.
In summary, we have described three methods (moments, percen­
tiles, and maximum likelihood), for estimating survival model parameters
directly from exact times of death, and have illustrated these methods for
one-parameter models (uniform and exponential), and for two-parameter
models (gamma, Gompertz, and Weibull). We also explored estimation of
one-parameter models from values of S0 (t;) using least squares.
8.2.2 Grouped Times ofDeath
This study design was described in Section 4.4, along with techniques for
estimating a tabular survival model from such data. We are now interested
in the estimation of parametric models, and we will consider the methods of
maximum likelihood and least squares for this purpose.
Suppose n persons are alive at t = 0, and we observe their deaths in
k non-overlapping intervals of equal length. Let d; be the number of deaths
observed in (i, i+ 1 ]. The probability of death in (i, i+ l] for a person alive at
t = 0 is Mo = S(i) - S(i+1), so the contribution of the (i+ 1y1 interval to the
likelihood is
L; = (S(i)-S(i+l)]d'.
(8.13)
The overall likelihood is
k-1
L = IJ [S(i) - S(i+ l)t·.
i=O
(8.14)
Then S(i) - S(i+ 1) is written in terms of the unknown parameters of the
chosen parametric model, and the parameters are found by maximizing L.
If the chosen S(t) is sufficiently simple, such as a one-parameter
uniform or exponential model, then convenient expressions for the MLE's of
the parameters can be found. Otherwise, as frequently occurs, the likelihood
equations must be solved numerically, or the likelihood is maximized nu­
merically without taking derivatives.
Univariate Models, Complete Data
191
[}:XAMPLE 8.71J Fit an exponential model to the grouped data of Example
4.4, estimating >. by maximum likelihood.
�OLUTION! In general, S(i) - S(i+1) = e->.i - e->.(i+l) = e->.;(1-e->.) .
Then L =
d + ln(l-e->.). L d;.
rr (e->.'(1-e->.)} ', so lnL = ->..I>.
�
�
�
4
_
4
d
4
i
4
= 0,
Next
I;i-d;
or
4
i=O
4
I:d,+I;i-d,
i=O
4
Since
i=O
4
do= 2, di = 3, d2 = 8, d3 = 6, d4 = 1, we find L d; = 20 and Li· d; = 41,
which produces>. = .39730.
i=O
i=O
0
For a two-parameter Weibull distribution, the probability of death in
(i,i+l] is S(i)-S(i+l) = exp[ - n!l · in+I ] - exp[ - n! l · (i+ll+1 ],
so from (8.14) ,
L
{
= TI
k-1
i=O
exp[-
_k_. ;n+I
n+1
] - exp[- _k_. (i+l)n+ IJ
n+ I
}d,
(8.15)
We can see that 81il = 0 and 01:L = 0 will not produce equations that are
easily solved for� and k. Since numerical computer routines are in existence
to directly find the � and k which maximize L, there is no need to take the
partial derivatives of ln L.
We recognize that S(i) - S(i+ l) = S(i) · q; is the multinomial prob­
ability that a person in the original sample of n at t = 0 will die in (i, i+ l ].
Then
E [D;] = n[S(i) - S(i+l)]
(8.16)
is the expected number of deaths in (i, i+l]. Since d; is the actual number of
such deaths, then least-squares estimates of the parameters of S(t) can be
found by minimizing
k-1
2
SS= I::{n(S(i) -S(i+l)]-d;}.
i=O
(8.17)
Estimation of Parametric Survival Models
192
!!EXAMPLE 8.8ij From the grouped data of Example 4.4, estimate the uni­
fonn distribution parameter w by unweighted least squares.
4
I
2
!SOLUTION! n = 20 and S(i)-S(i+l ) = -z:;, so SS = L [ 20
�-d;] ,
4
i=O
20
20 = 0, which yields fv = 5
leading to ddSS
[
)
]
(
2
L
=
.
·
� - d; · - w 2
w
1=0
I\
I\
I\
(The results of Example 4.4 for 2lq0 · S(3) and q3 could be compared with
those produced by the parametric models estimated m Examples 8.7 and
8.8.)
0
As an alternative to minimizing the total squared deviations of actual
from expected grouped deaths, we could approximate the hazard rate at the
midpoint of each interval (see page 198), and define SS to be the sum of
squared deviations of (approximate) observed hazard from the parametric
model hazard, in terms of the unknown parameters, of course, That is
(8.18)
from which the least-squares estimates of the A(t) parameters can be found.
This is frequently easier to do than the grouped deaths approach considered
above, since A(i) is frequently a simpler function of the parameters than is
S(i)-S(i+ l ).
The hazard rate approach will be considered further in Section 8.3.2.
8.3 UNIVARIATE MODELS, INCOMPLETE DATA
We now explore the estimation of parametric survival models in the pres­
ence of incomplete data. We will consider two estimation approaches, name­
ly, the methods of maximum likelihood and least squares. The former will
be used in cases where we estimate our parameters directly from the data.
The latter will be used where we .first process the basic data to obtain esti­
mates qx (or more likely a constant µ) for the interval (x, x+ 1 ], as described
in Chapters 6 and 7, and then estimate our model parameters by fitting our
model to this sequence of initial estimates.
8.3.J Maximum Likelihood Approaches
As a first example, consider a sample of n laboratory mice all alive at t = 0.
We observe the exact time of each death up to time t = r, and cease
Univariate Models, Incomplete Data
193
observation at that time with some of the mice still alive. Since not all have
died we have an incomplete data situation. Note that the study design is
longitudinal (not cross-sectional), and those that are not observed to die are
enders, not random withdrawals. In a clinical setting this type of study is said
to be truncated.
The contribution of each death to the likelihood is the PDF for death
time
that death actually occurs. The contribution for each survivor at
the
at
t = r is simply the probability of living to t = r, namely S(r). If there are d
deaths in total, out of the original sample of size n, then we have
d
l = [S(r)r-d · ITf(t;),
i=I
(8.19)
where /; is the time of the i'h death.
p:XAMPLE 8.91! Suppose the study described in Example 8.2 had been
truncated at t = 9. Estimate the exponential parameter ). in this case.
[SOLUTION! At t = 9, 6 deaths have occurred and 4 are still alive, so we
6
have L = [S(9)]4 • ITJ(t;). In this case, S(9) = e-9>. and /(t;) = >.e->.t;,
i=I
I,,.= 3 '4'5'7 '7 '8 ' giving us L = >..6 • e-34>. • e-36>. = ).6 • e-7o>.. This produces
dlnL
6 -70 =
"
6
= x
0, and)..=
lnL = 6 · ln>..-70>.., so�
70 = .08571.
D
More generally, suppose the i1h individual in our study comes under
observation at time r; and leaves observation at time t;. An individual can
leave observation by death or by the ending of the observation period (i.e., as
a scheduled ender), but we assume, for now, that there are no random
withdrawals. In other words, death is the only random event that can occur,
so we are operating in a single-decrement environment.
.
.
S(t;)
. tirom r; to I; IS
1 tty of 1·1vmg
The probab·1·
t;-r;Pr; = S(r;), and th"IS IS
the total contribution to the likelihood for those alive at r; who cease
observation at t; without dying. For the deaths, we need the PDF at time t;,
so 1,-rPr, must be multiplied by)..(/;). Thus we have
.
l = IT S(t;) >.(t;) . II S(t;)'
S(r;)
S(r;)
v
15
(8.20)
where the first product includes all deaths and the second includes all who
exit alive.
194
Estimation of Parametric Survival Models
tains
Recall the indicator variable D; defined by (7. l 7). Since (8.20) con�
for all n persons in our study, and contains >.(t;) only for the
f�;,�
deaths, we can write the likelihood as
L =
. (,\(t;)]°;
TIi==I S(t;) S(r;)
(8.21)
Furthermore, since >.(t;) = f(t;)/S(t;), we could write
1
°
[S(t;)] -o; [f(t;)] i
L = TI
S(r;)
.
i=I
(8.22)
The reader should be able to see that (8.20), (8.21) and (8.22) are equivalent
expressions.
j!EXAMPLE 8.10!! Consider the following six patients who received artifi.
cial hearts on the dates shown. Four died by December 31, 1996.
Patient
I
2
3
4
5
6
Date of
Date of
Death
Transplant
Jananuary I, 1995 April 1, 1996
April I, 1995
April 1, 1996
July 1, 1995
October 1, 1995
July I, 1996
Jananuary l, 1996
October I, 1996
April 1, 1996
Estimate the exponential parameter >. from the data of this sample, using cal­
endar year 1996 as the observation period.
{soLUTION] At time of entry into the study, the six patients have the fol­
lowing values of r;: r1 = I, r2 = . 75, r3 = . 50, r4 = . 25, rs = 0, r6 = 0. At
time of exit from the study, the values oft; are as follows: t 1 = 1.25, t2 = I,
t3 = 1.50, t4 = .75, ts= I, t6 = .50. (Note that t3 and ts are ending times,
and the other t; are death times.) From (8.21) we find the log-likelihood
In L = I:(-,\· t; +>. · r; +In>.· o;]. When d��L is set to zero, we obtain
i== I
Univariate Models, Incomplete Data
195
6
LD;
"
.X -
6
= 35 = 1.1429. The formula for >. is similar to those
I\
4
i=l
L(t;-r,)
i=I
obtained in Chapter 7 for MLE's under the exponential assumption, because,
in both cases, ,,\ is assumed to be constant throughout the study interval. D
Now suppose withdrawal is a random event along with death, so that
we are in a double-decrement environment. The probability of living from r;
to I; is now called
(r) _
11 -r;Pr;
-
t(d)
l;-r;Pr;
t(w)
· t1-r;Pr,
,
(8.23)
as introduced in Section 5.3, if death and withdrawal are independent. In
terms of survival distribution notation, let Sd(t) denote "survival against
death" and let Sw (t) denote "survival against withdrawal." Then
(r _ Sd(t;) · Sw (t;)
°
t;-rPr; ) - S ( ;) S
· w (r;)
d r
(8.24)
Further, let >.d(t) = µ�d) and >.w (t) = µ�w) denote the hazard rates (forces) of
death and withdrawal, respectively.
In this situation l; is the time of death, random withdrawal, or sched­
uled ending from the sample, recalling that scheduled ending is not a random
event.
Clearly all three categories of exit will each contribute t;-r;Pt) to the
likelihood. As well, each death will contribute >.d (t;) and each withdrawal
will contribute Aw (t;). Thus we have
(8.25)
(8.26)
Let us introduce a second indicator variable, 'Yi, where
_
'Yi - {
I
if person i is a random withdrawal
O otherwise
(8.27)
Estimation of Parametric Survival Models
196
Then in terms of survival distribution notation, our likelihood can be written
as
(8.28)
Now the survival model for death depends on certain parameters, say
8 1 and 82 . Similarly the model for random withdrawal depends on certain
parameters, say ¢ 1 and ¢2 . If the 0-parameters and the ¢-parameters are
mathematically independent of each other, then the terms Sw (t) and Aw(!) are
multiplicative constants when we estimate the 0-parameters, and so can be
ignored. Similarly Sd(t) and .\ d(t) can be ignored when we estimate the
¢-parameters. In other words, when we estimate the 0-parameters, (8.28)
produces the same results as does (8.21) under the independence assumption.
If the independence assumption is not valid, then the estimation of
parameters is much more complex. We will not consider such situations in
this text.
!!EXAMPLE 8.111! Suppose Patient 4 in Example 8.10 withdrew on July I,
1996, instead of dying. Suppose mortality follows an exponential model and
withdrawal follows a uniform model. Estimate .\ and w, assuming indepen­
dence.
!SOLUTION! We first note that Sd(t) = e->.,, .\d(t) = .\, Sw (t) = u1: t , and
Aw (t) = w�r From (8.28), we find the log-likelihood to be
InL =
so that
L [lnSd(ti) + lnSw(t;) - lnS (r;) - lnS (r;)
6
d
i=l
w
+ D; · ln..Xd(t;) + "(; · ln.).,.(t;)],
�[ d
d
d
�- = LJ d>.. lnSd (t;) - d>.. lnSd(r;) + D; · d>. In Ad (t;)] , smce
.
a In L
t=I
%>.. Sw (t) = %>.. Aw (t) = 0. This yields the same likelihood equation as in Ex6
ample 8.10, except that here we have E D; = 3 rather than 4. Thus we have
i=I
8
� = 33., = .85714. Similarly, 8 w Sd(t) and 88uJ >.d(t) are zero, so the deriva-
81nL = O I eads, a fter some s1mp
. equation
. aw
. 1·fi
. to
ttve
1 1cat1on,
.J
Univariate Models, Incomplete Data
197
where the last term results from 14 = 1 and t4 =. 75. This equation must
be solved for D by numerical methods.
D
8.3.2 Least-Squares Approaches
Let us suppose that one of the estimation procedures of Chapters 6 and 7,
actuarial, moment, or maximum likelihood, has been used to produce a se­
quence of qx for each estimation interval (x, x+ 1]. From 'tJx we obtain an
estimate of the force of mortality at x+ the midpoint of (x, x+1], from
f,
(8.29)
by making the exponential assumption for fx+s , 0 < s < 1, or from
I - -21 ·qI\x
)
(8.30)
by making either the uniform or hyperbolic assumption. Of course, if we use
Estimator (7.23), a full data estimator under the exponential assumption,
then we obtain 'ilx+ 112 directly.
We now wish to fit this sequence of µx+i;2, from Chapter 6 or 7, to a
chosen functional form, such as Gompertz, Weibull or Makeham. Once the
chosen model's parameters have been estimated, then we can calculate our
estimate of µx+i;2 from this parametric survival model. Since we will want
to reserve the notation µx+t/Z for such an estimate, the µx.+t/2 result from the
Chapter 6 or 7 approach must be renamed to avoid confusion. Let us denote
such preliminary estimate by µi+i;2, where the superscript means "observed." Thus "o" denotes an estimate calculated directly from the sample
data, and the familiar """ denotes an estimate obtained from the parametric
model fitted to the µ;+t/2 sequence.
To fit a Gompertz model to the µi+i/2 sequence, we recall, from
Equation (2.31), that µx = >.(x) = Be\ which is not linear in the unknowns
B and c. However, the transformation
In µx
= In B + x · In c = a. + {3x
(8.31)
is linear in a. and {3. If the domain of µi+i/2 is from x = a to x = b, then we
have
198
Estimation of Parametric Survival Models
SS =
,e(
b
?
� Wx [a+ x+!) - lnµ:+1/2 r
x=a
(8.32)
as the general weighted sum of squares to be minimized by the least-squares
" "
estimates a and ,6.
It should be mentioned that the estimates of B and c found from
B = e0
I\
A
and � = ef3
A
(8.33)
will be somewhat different from those found by minimizing the sum of the
squared differences between Bc x+1!2 and µ:+112 , without making the log
transformation.
If Wx = I for all x we have unweighted least-squares estimation.
Some evidence exists that the unweighted approach gives better estimates of
the true parameters (see, for example, Gehan and Siddiqui [31]), and it is
certainly easier to apply.
!!EXAMPLE 8.12!! Estimate the Gompertz parameters Band c by an un­
weighted least-squares fit to the following central rates observed from sam­
ple data.
x
31
32
33
34
35
mxo
.02639
.02914
.03557
.04121
.04966
ffiQ.LUTfoNl An observed central rate over (x, x+ l] is the same as the
(constant) force over (x,x+ 1) under the exponential assumption. Thus we
take µ�+l/2 = mi. The least-squares equations, which result from differentiating (8.32) with respect to a and ,6, respectively, are
5.00 ·a:+ 167.50 · ,6 =
- 16.69830
and
167.50 · o: + 5621.25 · ,6 -- - 557.78192,
Univariate Models, Incomplete Data
199
/\
which solve for et= -8.73694 and /3 = .16111. This in tum produces
/\
/\
0
the estimates B = .00016054 and c = 1.17482.
/\
To fit a Weibull model, we again find that µx = >.(x) = k· x11 , from
(2.35), is not linear in k and n. Again the log transformation
In µx
= Ink + n· ln x = a + fi· In x
(8.34)
is linear in a and /3, so we have
(8.35)
as the general weighted sum of squares to be minimized by the least-squares
/\
estimates a and /3. Then estimates of k and n are obtained from
/\
k = e0
/\
A
and n = (3.
/\
/\
(8.36)
Fitting a Makeham model, with µx = >.(x) = A+Bc X , from (2.33),
is a greater challenge, since the log transformation ln µx does not produce a
linear function in the parameters to be estimated. The following approach,
which was suggested by Nesselle [58], might be used.
We begin by approximating µx = A+Bc x by a Taylor series. Let
µ
µx
µ
0
µx = µx +(A-A o )· oaA ' o +(B-Bo )· oasx' o + (c-co)· oacxl o+··· .
(8.37)
The method requires initial estimates ofA, B, c, which we call Ao, Bo, c0. In
(8.37) the notation lo means that the indicated partial derivative of µx is
evaluated at Ao, Bo, co. Since µx =A+ Be x , the indicated partials are
oµx _ 1
oA - '
8 µx
c
o
= Bxc x-l
(8.38)
Letting A -Ao = dA, B - Bo = dB, c - Co = d e , we have
'°"
2
l x-1/2
x+l/2 + d
0]
(''S = � Wx [o
- µx+
o,
µx+l /2 + d
A+ d
B. Co
c. B(
0 x+ 2)CO
l/2
x=a
(8.39)
b
200
Estimation of Parametric Survival Models
as the general weighted sum of squares to be minimized by the least-squares
·
/\
/\
/\
x+ l /2
O
estimates dA, ds, de. Note that µx+l
, whereas µi+I/2 is
/2 = Ao+Boc0
the value observed from sample data. The derivatives of SS with respect to
/\
/\
/\
dA, ds and de are equated to zero to produce d A , ds, d e , which are added to
Ao, Bo, co, respectively, to produce A 1 , B 1 , c 1 , say. Now these last values
could be taken as our estimates of A, B, c, but they are not the true Ieast­
squares estimates since (8.37) is only an approximation. We consider A 1 , B 1 ,
c1 to be improved estimates from A0 , Bo, c0 , and use them in place of Ao,
/\
/\
/\
Bo, co in (8.39) to solve again for d A , ds, d e and hence other estimates called
A2, B2, c2. Continued use of this iterative approach will show a convergence
/\ /\
of the successive estimates to the least-squares estimates A,B).
8.4 HYPOTHESIS TESTING OF PARAMETRIC MODELS
We now wish to explore briefly the area of formal statistical tests that can be
performed to determine the acceptability of a fitted parametric model as an
adequate representation of the true underlying model. We wish to do this
first for the case of grouped times of death and then for the case of exact
times of death. In both cases we are assuming the complete data situation of
Chapter 4.
8.4.1 Grouped Times of Death
Suppose, in the grouped times of death situation of Section 8.2.2, the param­
eters of the chosen parametric model have been estimated by some procedure
such as maximum likelihood. We are then hypothesizing that the fitted mod/\
el, S(t), is a good representation of the real S(t), and we wish to test this
hypothesis. Let
E;
/\
/\
n[ S(i) - S(i+ 1 )]
(8.40)
represent the expected number of deaths in (i, i+ l] according to the hypoth­
esized model. Recall that d; is the number observed in (i, i + l] from our
sample. The quantity
2
x =
I:
(£;--;_�;)
E;
i=O
k-1
A
2
(8.4 l)
201
Hypothesis Testing ofParametric Models
2
has approximately a x distribution with k- 1-r degrees of freedom, where r
is the number of parameters in S(t) which were estimated from the data. If
the hypothesized model is fully specified including its parameters (i.e., if the
parameters are not estimated from the data), then the approximate x2 statis­
tic defined by (8.41) has k-1 degrees of freedom.
(lt:XAMPLE 8.13lj Test the hypothesis that the S(t) determined in Example
8.7 is a good representation of S(t).
E;
[e--3973i( l -e-·3973)], for
= 20
�OLUTIONI Since � = .3973, then
A
A
A
i = 0, . . . ,4, producing the values E0 = 6.5574, £1 = 4.4074, E2 = 2.9624,
A
A
A
E3 = 1.9911, £4 = 1.3383. In addition, Es+ , the expected number of deaths
in (5, oo), is 20 · e- 5,\ = 2.7435, and the number observed is d5+ = 0. Using
(8.41) with k = 6, we find x2 = 23.0839. Since only one parameter was
estimated, then x2 has 6-1-1 = 4 degrees of freedom. A table of x2 values
shows that the probability of obtaining a departure measure of 23.0839 or
A
more is less than .001, so we find very significant evidence against our S(t)
D
as a representation of S(t).
Since a small x2 value indicates a good fit of S(t) to the data, one
might determine the parameters of S(t) so as to minimize (8.41). Parameters
determined in this way are called the minimum x2 estimators of the param­
eters.
Since the unknown parameters would appear in both the numerator
and denominator of (8.41), the minimization mathematics could be quite
complex. As an alternative, we might find the parameters by minimizing
k I
2
(E; - d;)
x,2 = �
�
d;
A
i=O
(8.42)
instead of (8.41). Parameters determined this way are called the minimum
modified x2 estimators of the parameters. Note that (8.42) is a weighted
version of (8.17), with weights of w; =
As the sample size approaches
infinity, minimization of (8.41) and (8.42) will produce the same parameter
estimates.
£.
!!EXAMPLE 8.14lj Repeat Example 8.8 using W; =
modified x2 estimator of w .
£ to find the minimum
202
Estimation of Parametric Survival Models
4
2
[soLUTIONI
1 [ 20
]
We have SS= L d: -z:; -- d; , so the denvat1ve equation
dSS
I\
� 1
i=O
•
.
I
w = 0 leads to w = 4 · L, d: = 8.50.
T
i=O I
0
8.4.2 Exact Times ofDeath
In this case we wish to compare our S(t), which is hypothesized as a representation of S(t), with the observed S0 (t). To do so we need a measure of the
departure of S(t) from S0 (t).
A fairly simple departure measure is the Kolmogorov-Smirnov sta­
tistic, defined as
Dn = max IS(t) - S0 (t)I ,
I
(8.43)
the largest absolute deviation between S(t) and S0(t) to be found over the
domain oft. The subscript of D11 reminds us that this measure depends on
the sample size n. We then calculate
(8.44)
as the actual departure measure to be tested.
For small n, tables of critical values of y for various significance
levels can be found in some books of tables or textbooks (see, for example,
Hollander and Wolfe [38]). As n --too, the probabilities can be calculated
from
Pr(Y> y)
2
L <-1>.i+l e-2fy-,
.i=I
(8.45)
so, in effect, (8.45) gives approximate probabilities for large values of n.
Eifill
Let us hypothesize that S(t) has the uniform distribution
!!EXAMPLE
S(t) = 1 - / for the sample of Example 8.2. Test this hypothesis using the
5
Kolmogorov-Smirnov statistic.
Hypothesis Testing ofParametric Models
203
�OLUTIONI Figure 8 .2 shows S0 (t) for that data, along with the hypothesized S(t).
1.00
.90
�S'(I)
.80
.70
.60
.50
.40
.30
.20
.10
3
0
4
7
5
8
10
12
15
!FIGURE 8.21
Table 8.3 compares values of S(t) with values of S0 (t), where S0 (t-) denotes
the value of S0 (t) just before t, and S°(t + ) denotes the value just after t.
TABLE 8.3
t
S(t)
3
4
5
7
8
10
12
15
.80000
.73333
.66667
.53333
.46667
.33333
.20000
0.00000
S0 (r) soc,+)
1.00
.90
.80
.70
.50
.40
.to
0.00
.90
.80
.70
.50
.40
.10
0.00
-
max
ls(t) - 8° (1)1
.20000
.16667
.13333
.16667
.06667
.23333
.20000
0.00000
From the table we see that the maximum deviation is .23333. Then we have
y = (.23333) ·
= .73786, and we seek Pr(Y > . 73786). From (8.45)
Jw
204
Estimation of Parametric Survival Models
2
2
2
we obtain 2 [e-2(7 3786) -e-8(7 3786) + e-18(.73786) _ --·] = .64762. (The first
1
omitted term in the summation is - e-32C·73786l , which is - 2.71 x 10-s.)
/\
Thus we find no significant evidence against the hypothesis of S(t) = 1- -t
1 5,
although our result here is suspect since we have only n = 10. Use of (8.45)
for Pr(Y > y) should probably be restricted to cases where n 2'. 50, say. O
In Example 8.15, S(t) = 1- / was fully specified, including the val­
5
ue of w = 15, without regard to the data. If we only assume the type of
parametric model, and estimate its parameters from the data, then the
departure measure Y = /n · Dn has different probability values than in the
case above. Furthermore, Stephens [59] has suggested that the departure
measure itself should be modified in such cases. For example, if an exponen­
tial model is suggested, with ,\ estimated from the data, then
(8.46 )
is a better measure to use than the unmodified Y =
Jn · D .
n
�XAMPLE 8.16!! Evaluate (8.46) for the exponential S(t), found by the
method of moments in Example 8.2, compared to the data of the sample.
l§.Ql,UTIONl Our hypothesized model is S(t) = e-·13157'. Values of S(t) are
compared with values of S 0 (t) in Table 8.4, where S 0 (r) and S 0(t+ ) were
defined in Example 8.15.
TABLE 8.4
t
S(t)
S0 (r)
S0 (t+ )
3
4
5
7
8
10
12
>12
.67387
.59079
.51796
.39812
.34904
.26828
.20621
<.20621
l.00
.90
.80
.70
.50
.40
.10
0.00
.90
.80
.70
.50
.40
.10
0.00
0.00
max
Is<,> - s (t)I
0
.32613
.30921
.28204
.30188
.15096
.16828
.20621
<.20621
Hypothesis Testing ofParametric Models
205
The maximum deviation occurs at t = 3-, so we have D 10 = .32613. Then
( JTo+.26+
y = (.32613= 1.09607. Stephens also found that
-�g)
}&)
the critical value y = 1.094 corresponds to the .05 significance level. Since
/\
our y = 1.09607 is very close to 1.094, we could reject the hypothesized S(t)
with approximately 95% confidence.
D
Recall that the Kolmogorov-Smimov test is based simply on the
/\
largest deviation between S(t) and S0 (t). A better measure of the departure
A
between S(t) and S0 (t) is given by the A 2 statistic, proposed by Anderson
and Darling [4], which is the weighted expected squared deviation of S(t)
from S0(t), where the deviations are weighted by the inverse variance of
S0(t). That is
2_
n
/\ 2
· [S°(t) - S(t)] -j(t) dt,
.
A - 100
0 S(t) F (t)
(8.47)
F
since, from (4.4b), Var[S0 (t)] = S(t)� (t)' andf(t) is the PDF for death at
time t. Of course f(t), S(t) and F(t) are not known; since S(t) is being hy/\
/\
pothesized as representing S(t), then (8.47) is evaluated by using S(t), F(t).
andf(t) in place of S(t), F(t) andf(t). Furthermore, sincej(t) = 0 for t< 0,
Thus we have
then we can use 00 in place of
ft.
J�
oe
/\
/\
/\
/\
A2 = n J [S(t) · F(t)r 1 • [S0 (t) - S(t)]2 · f(t)dJ.
-oc
(8.48)
Finally, it can be shown (see Appendix C) that (8.48) is equivalent to
A2 =
/\ n-i+d] } ,
- n - n}�
· L.)2i - 1) {A
In [F(t;) · S(t
(8.49)
i=I
which is a more convenient formula for calculating A2 than is (8.48).
!!EXAMPLE 8.17!! Let us hypothesize S(t)
ample 8.2. Calculate the A2 statistic.
= e-.os, for the sample of Ex­
206
Estimation of Parametric Survival Models
!SOLUTION! Table 8.5 shows the evaluation of the summation in (8.49).
TABLE8.5
i
1
2
J
4
5
6
7
8
9
10
/\
/\
f;
F(t;)
S(tu-;)
3
4
5
7
7
8
10
10
10
12
.13929
.18126
.22120
.29531
.29531
.32967
.39346
.39346
.39346
.45118
.54881
.60653
.60653
.60653
.67032
.70469
.70469
.77880
.81873
.86070
(2i-t){ In [F(t;) · S(t11-;)]}
-
2.57120
- 6.62347
10.04345
12.03812
- 14.57757
- 16.05626
- 16.67605
- 17.74165
- 19.25721
- 17.97207
-- 133.55705
-
-
Then A2 = - 10- 1�(-133.55705) = 3.35571. Stephens [72] found that
y = 3.070 corresponds to the .025 significance level, and y = 3.857 corres­
ponds to the .01 level for this A 2 statistic, so our y = 3 .356 corresponds
roughly to the .019 level.
0
If the hypothesized S(t) has parameters estimated from the data, then
the A 2 statistic should be modified. For example, an exponential model with
). estimated from the data should use the modified measure
y = A 2 (1 +
·to) '
(8.50)
according to Stephens [73].
8.5 CONCOMITANT VARIABLES IN PARAMETRIC MODELS
Throughout most of this text we have only considered survival models that
were a function of chronological age, S(x), or those that were a function of
time since some initial event, S(t). In both cases the model was univariate.
Many cases arise in which survival probabilities are a function of
two or more variables, such as those used for insurance premium calcula­
tions which depend on age at issue as well as time since issue. This leads us
Concomitant Variables in Parametric Models
207
to the select survival model S(t;x) defined in Section 1.2.1. The tabular form
of S(t;x) was described in Section 3.6, and illustrated in Table 3.5. There we
saw that the concomitant variable age at issue, denoted [x], was taken into
account by having a separate S(t;.x) for that value of x. In other words, each
row in a select table constitutes a separate univariate model (varying with t
only); age at selection is reflected by choice of the appropriate row. For this
reason we say that the concomitant variable has been taken into account
through separation.
Similarly, suppose we consider the survival of cancer patients as a
function of time since diagnosis. We might believe that type of cancer, sex
of the patient, and type of treatment all affect survival, so we would estimate
a separate S(t) for each type/sex/treatment combination. Again S(t) is uni­
variate, with concomitant variables taken into account by separation.
We now wish to consider parametric models, in which survival
probabilities are determined as a function of both time and the accepted
concomitant variables. In this type of model the concomitant variables have
been taken into account by inclusion, rather that by separation.
The functional form for the multivariate parametric model should
allow the variables to interact in a logical manner, a priori to testing a pro­
posed model vis-a-vis sample data. That is, the model should be plausible in
light of our knowledge of physiology, gerontology, and so forth. In this text
we will not be concerned with the question of model appropriateness, but
rather with the mathematical question of estimating the parameters of such
models from sample data.
8.5.1 Bivariate (Select) Models
First let us explore the idea of estimating the parameters of a bivariate para­
metric model S(t;x), where x is the age at selection and t is duration (time)
since selection. (Note that if different S(t;x) are estimated for males and
females, then sex is accounted for by separation. Allowing for sex by
inclusion will be considered in the following section.)
As mentioned above, a logical form for S(t;.x) will depend on the
nature of the aging process, the nature of the selection process, and the inter­
action between the two. This has been studied by Tenenbein and Vanderhoof
[74], who then proposed the forms for S(t;x) which we will consider in this
section.
As mentioned earlier in this chapter, it is frequently easier to work
with the hazard rate function than with the survival function for purposes of
parameter estimation. Thus we will consider
Estimation of Parametric Survival Models
208
J\(t;x) =
- � In S(t;x),
(8.51)
which, in actuarial notation, is denoted by µ[xJ+1
Consider the model
_ Br t c x+t '
µ(x]+t -
(8.52)
where x+t is the attained age. To estimate B, r and c, we assume that we
have values of µ[.TJ +i+I/2 from sample data, and we will fit (8.52) to these
values by least squares.
As in Section 8.3.2, we will use a log transformation and actually fit
In µ[x]+t = In B + t · In r + ( x+t) · Inc = a+ f]t + 1(x+t)
(8.53)
to the values of In µ[x +t+l /i· Thus
]
SS -
LL Wx, [a+ f3(t+!) + ,(x+t+!) - In µf J+t+l/i] (8.54)
b
k-1
t
x=a t=O
x
2
is the general weighted sum of squares to be minimized by the least-squares
estimates &, /3 and ,9, where the select period is k years.
/\
The three equations which solve for &, /3 and ,9 are found by setting
· l
ass ass , and ass to zero. T0 Slmp
· .. 1et Ux,t = 1n µo[x}+t+l/2.
"Ify notation,
a,
aa, a/3
Then our least-squares equations are
+ 1 · EE(x+t+!)wx,r
= EEwx,tUx,r
+ 1 · EE(t+!) (x+t+!)wx,r
= EE(t+!)wx,1Ux,t
a· EE(x+t+f)wx,t + f3 · EE(t+t) (x+t+f)wx,1 +
where EE denotes I:
b
k-1
L·
x=a t=O
"{ · EE(x+t+f)2 wx,r
= EE(x+t+!)wx,1Ux,1
(8.55)
Concomitant Variables in Parametric Models
209
l!j:XAMPLE 8.18!\ Let a= 40, b = 42, and k = 2. Find the coefficients of
a, /3, "I in the equation which results from a8 = 0, in an unweighted least
5/
squa res approach.
�LUTION! For a, we have
LI: ( +!) = L (!+�) =
42
I
x=40 1=0
For /3, we have
1
42
x=40
6.
Finally, for "I we have
As a second example of a bivariate model, consider the model
l/(t+l}
µ[x]+t = Bcx+t (Hdx+t)
(8.56)
The log transformation gives
= In B + (x+t) · Inc + ( i! 1) · In H + ( :+{) · Ind
Inµ[xJ+t
= a + f3(x+t) + ,( t! 1 ) + o(:_-tf),
and �, /3,
/\
t o are found so as to minimize
/\
where Ux,t = lnµr.�J+i+l/2.
(8.57)
210
Estimatioll of Parametric Survival Model,
8.5.2 General Multivariate Models
Now let us consider the general case where there are s concomitant vari­
ables, z1, z2, ···,Zs, We will use z to denote the column vector of concom­
itant variables, and will denote the survival function by S(t;z) and the hazard
function by ,\(t;z). As usual,
so
and
- 3t lnS(t;z),
,\(t;z) =
(8.59)
S(t;z) - exp (- J.' A(u;z)du) ,
(8.60)
f(t;z) = S(t;z) · ,\(t;z).
(8.61)
Each person in the group to which S(t;z) applies will have his or her
own hazard rate, wherein the values of z apply specifically to that person.
For example, suppose z1 is age at selection, and z2 and z3 are indicator
variables defined by
and
0 if male
{
2
z =
I if female
Z3 = {
0 if smoker
I
if nonsmoker
(8.62)
.
(8.63)
If person i is a 40-year-old, nonsmoking female, then her hazard rate func­
tion is ,\(t;z;), where z� = [40, 1, 1].
In many cases, of which the above example is one, the concomitant
variables are factors which exist at time t - 0, and cio not change as I in­
creases. This concept of z being constant with respect to t is easier to deal
with, and is the only case we will consider in this text.
In addition to the variables t and z, a parametric S(t;z) will also
depend on a set of parameters. Just as we did for univariate models earlier in
this chapter, we wilt choose a parametric form for S(t;z), and then estimate
its parameters from sample data, using the method of maximum likelihood.
Concomitant Variables in Parametric Models
211
To construct the likelihood function, let us assume that person i
es
under
observation at time r; and leaves observation at time !;, either
m
co
alive or by death. We will assume that either withdrawals do not occur at
random, or, if they do, they are independent of mortality. In either case they
can be ignored in the likelihood function as we saw in Section 7.4.1. The
contribution to the likelihood by the i'h life is
f(t;;z;)
L; - S(r;;z;)
(8.64)
if that life is observed to die, or
S(t;;z;)
S(r;;z;)
(8.65)
if the life terminates observation alive at I;. Both cases can be incorporated
by writing
L; =
S(t;;z;) · [�(t1;z1)]61
S(r;,z;)
(8.66)
where 6; is the familiar indicator variable defined by (7 .17). The overall
likelihood is given by
(8.67)
In theory, we can always estimate the parameters in (8.67) so as to
maximize L, where the difficulty of the maximization will depend on the
complexity of >.(t;z), and thus of S(t;z). We now consider two models for
>.(t;z), and explore the maximization of (8.67) under each of them.
8.5.3 The Additive Model
In this model the total hazard >.(t;z) is made up of an underlying hazard
which is a function of time, >.(t), and additional hazards arising from the
concomitant variables, all added together. lf, in general,
(8.68)
represents the extra hazard at time t due to the J'h concomitant variable, then
the total hazard is
Estimation of Parametric Survival Models
212
,\(t;z) = >.(1)+
I: h (t). &<z ).
s
J=I
j
1
(8.69)
Now (8.68) is written to suggest the most general form for .>.(t-z·)
''J'
expressed as the product of a function of time and a function of zi . (By
subscripting the functions h and g with j, we allow them to vary with j.) As
a special case, we might have gj(z1) = z1 for all j. Similarly we might let
h;(t) = a1 , say. If these two simplifications are combined, we then have
(8. 70)
so thatthe additive model becomes
.>.(t;z) = .>.(t) +
L! a;zJ­
s
J=
(8.7 l)
As a final simplification of the additive model, we consider the underlying
hazard .A(t) to be a constant, say a0, which implies that the underlying sur­
vival function is exponential. Then by defining the dummy variable z0 = 1,
(8.71) can be written as
,\(t;z) =
-
I: a;z1 = a'z,
s
j,=,O
(8.72)
where a' = [ao, a1,···,as ] and z' = [zo,z1,···,z5]. Since the hazard is
constant, the overall survival function is exponential. Furthermore, since
(8. 72) is linear in z, this .>.(t;z) is sometimes called the linear-exponential
model.
To esiimate the parameters a of (8.72), we construct the likelihood
given by (8.67). Note that, for this case,
S(t;;z;)
--= exp ( - 1t; >.(u;z;)du) = exp[-atz;(t;-r;)],
r,·
S(r;;z;)
(8.73)
since ,\(u;z;} = a'z; is a constant. Then we have
n
L = TI ((a'z;)8; · exp[-a'z;(t;-r;)]),
i=I
(8.74)
Concomitant Variables in Parametric Models
213
which leads to
n
n
i=l
i=l
lnl = L<\-ln(a'zJ-L(ti-r;)(a'zi )-
(8.75)
finally, 8J� L = 0 leads to the equation
'}
(8.76)
where Zji is the j'h concomitant variable for person i. The (s+ I) equations of
this form, j = 0, 1,- · · ,s, are then solved simultaneously for tz0, fz 1 , - • ·,
which must usually be done by numerical methods.
:t
[EXAMPLE 8.1911 The survival of patients with a certain lung disorder is
believed to depend on age at onset and smoking habits, as well as time since
onset. Assume that a linear-exponential survival model is appropriate. Let
z1 be the age at onset, and z2 = 1 if a smoker, 0 if a nonsmoker. A sample of
six patients was observed from t = 0. Table 8.6 gives the values of z1, z2,
the times of death for those who died, and the survival times for those alive
when the study was terminated. Determine the equations to be solved for fi0 ,
/\ and a-i.
/\
a,,
Patient
I
2
3
4
5
6
Zt
50
52
53
55
58
60
TABLE8.6
Time of Time at Study
Termination
Death
Z2
I
6.0
4.8
0
2.2
0
9.2
I
1
8.0
5.5
0
!SOLUTION! Note that r; = 0 for all i, b; = I for i = I, 4, 6, and the a'z;
vector is a'z; = a0+a1z!i+a2z2i. (Recall that zoi = l for all i.) We first
calculate a'z; for i = 1,2,- · -,6. The results of this are a'z 1 = ao+50a, +a2;
a'z2 = ao+52ai (z2 = O); a'z3 = ao+53a1 (z2 = O); a'z.i = ao+55a1 +a2;
a'zs = ao+58a1+a2; a'Z<, = ao+60a1 (z2 = 0). Now for j = 0, (8.76) be­
comes
Estimation of Parametric Survival Models
214
1
ao + 50a1 + a2
I
I
- 3 7
+ ao + 55a1
+ a2 + ao + 60a1 � 5· '
since oi = 0 for i = 2, 3,5, and zo; = I for all i. Next, for j = I, (8. 76) be­
comes
50
ao + 50a1 + a2
60
55
+ ao + 55a1
+ a2 + a0 + 60a1 = 1966·2·
For j = 2, we have zji = 0 for i = 2,3, 6,and oi = 0 for i = 2,3,5. Thus the
first summation in (8.76) has nonzero terms only for i = 1 and i = 4,yield­
mg
I
+
I
= 23·2'
ao + 55a1 + a2
ao + 50a1 + a2
where only i = 1,4,5 contribute to the second summation.
0
8.5.4 The Multiplicative Model
In this model the underlying hazard and the additional hazards from the con�
comitant variables are multiplied together, rather than added.
The most popular model in this family is the Cox model (see Cox
[21 ] ),where
(8.77)
so the entire hazard is
(8.78)
Furthermore, if the underlying hazard is constant over time, then we can let
>.(t) = >. = eao , so that
>.(t;z) = exp
(t
J=O
ajZJ) = e•'z,
(8.79)
where a'= [ao, a1,···,as ],z' = [Zo,z1,···,z s ],and zo = l,necessarily.
The hazard rate is again constant, so the survival model is exponen­
tial. Furthermore, In >.(t;z) is linear in z, so (8.79) is sometimes called a
log-linear exponential model.
Co11comitant Variables in Parametric Models
215
To estimate the parameters a of (8.79), we construct the likelihood
given by (8.67). Since .\(t;z;) =
- ea'z; is constant, then
S(t;;Z;)
S(r;;z;)
= exp[ - (t;-r;) · ea'z'],
(8.80)
so that
n
L = Il ( ea'z,]o• . exp ( - (t;-r;). ea'z•J '
(8.81)
LO; · a'z; - L (t;-r;) · e 'z;.
(8.82)
i=I
leading to
lnL =
n
n
i=I
i=I
a
Finally, 8J� L = 0 leads to the equation
1
n
n
LO; · zji - L�;(t;-r;) · e•'z; =
- O.
i=I
i=I
(8.83)
Again we recognize that there are (s+ I) equations of this form, indexed by
j = 0,1,· · ·,s, which must be solved, usually numerically, for tt0 ,tt 1,· ··,tis.
!!EXAMPLE 8.20!1 Repeat Example 8.19 by assuming a log-linear expo­
nential model.
!SOLUTION! Recall that r; = 0 for all i, and D; = l for i = I, 4, 6. For
j = 0, Zji = zo; = l for all i, producing, for the derivative Equation (8.83),
3 - 6 · exp(ao+50a 1 +a2)- 4.8 · exp(ao+52a1) - · · · - 5.5 · exp(ao+60a,) = 0,
where the six values of a'z; were recorded in Example 8.19. For j = 1 we
have
(50+55+60) - (6)(50) · exp(ao+50a1+a2)- · · · - (5.5)(60) · exp(ao+60a1) = 0.
For j = 2, Zfi = z2; = 1 for i = I, 4, 5, so derivative Equation (8.83) be­
comes
2 - 6 · exp(ao+50a 1 +a2)- 9.2 · exp(ao+55a 1 +a2) - 8 · exp(ao+58a, +a2) = 0.
0
Estimation of Parametric Survival Models
216
8.6 SUMMARY
In this chapter we presented several approaches to the estimation of the
parameters in a chosen survival model from sample data, considering various
ways in which the data might be available (e.g., exact times of death,
grouped deaths, initial estimates of values of S(t) or .X(t), etc.). We also
explored our estimation problem under both complete and incomplete data.
Two major techniques were emphasized in this chapter, namely
maximum likelihood and least squares. Both are popular methods with avail­
able computational packages to handle the otherwise formidable (or even
prohibitive) number of calculations required to obtain numerical results in
many cases.
The reader should bear in mind that maximum likelihood estimates
are not guaranteed to result frorri the standard calculus approach to maximiz­
ing the likelihood function. The function may be unbounded (i.e., a maxi­
mum does not exist), or it could have several local maxima.
We have given only a brief treatment of hypothesis testing and mo­
dels with concomitant variables. Many other texts are available which give a
complete treatment of these topics.
The general concomitant variable model would appear to have a na­
tural life insurance application. In addition to the obvious concomitant vari­
able of age at issue, already incorporated into the select S(t;x) model, other
concomitant variables (or covariates) such as sex, smoker/nonsmoker, blood
pressure, weight, and other underwriting factors could be incorporated into a
parametric hazard function applicable to the particular insurance applicant.
From this personal hazard function, the insurance premium and other policy
values could be calculated specifically for this applicant. This interesting
possibility has been explored by Brown and Brown [17].
8. 7 EXERCISES
8.1 Introduction;
8.2 Univariate Models, Complete Data
8-1.
Find the asymptotic variance of Estimator (8.2).
8-2.
Since ± is the mean of T, the time of death, then ,\ -1 = (, say, is the
average time at death of the exponential survival model. Consider the
sample of independent exact times of death ti, i = 1,2, · · · ,n.
(a) Estimate ( from this sample data by the method of moments.
/\
(b) Show that ( is an unbiased estimate of (.
/\
/\
(c) Find the variance of(. (N_ote that this is Var(() for finite n as
well as the asymptotic variance.)
Exercises
217
8-3.
In an experiment involving 6 newborn mice, the following exact times
of death were observed: 0.6, 2.2, 2.3, 3.1, 4.6, 7.2. An exponential
survival model is to be fitted to the data. Determine the MLE of>..
8-4.
For a group offive impaired individuals followed from birth, the force
of mortality is assumed to be a linear function of age such that
µx = bx. Ifthe deaths occurred at exact ages 1, 2, 3, 4 and 5, find the
MLE ofb.
8-5.
Using the sample data of Example 8.2, estimate the parameter w ofthe
uniform distribution (a) by the method of moments, and (b) by the
method of medians.
8-6.
The survival times of IO persons, each age 7 5 at diagnosis ofa disease,
were 0.5, 1.3, 1.4, 1.6, 1.7, 2.2, 2.9, 3.8, 3.9 and 4.2 years. Using the
method of moments, a gamma distribution (see Example 8.3) with
a= 2 is fit to the data. Find the moment estimator of{3.
8-7.
Verify the numerical results ofExample 8.4.
8-8.
Verify the numerical results ofExample 8.6.
8-9
In a study of 12 individuals from birth, the. following times of death
(in years) are observed: 0.38, 0.82, 1.23, 1.61, l .86, 2.20, 2.70, 2.95,
3.47, 4.25, 5.16, 5.99. The Nelson-Aalen method is to be used to
obtain approximate values of A(t). We then wish to fit the data to an
extreme value distribution for which A(t) = exp [ln (Rla)+at], by
matching the fitted values of A(t) to the Nelson-Aalen approximate
values at the times of the third and ninth deaths. What is the resulting
estimate ofa?
8-10. Two different studies each have a sample of 3 persons alive at t = 0.
For Study I, two persons die in (0,1] and one person dies in (1,2).
For Study 2, one person dies in (0,1] and two persons die in (1,2].
For Study 1 we assume an exponential model with parameter >.1, and
for Study 2 we assume an e1ponenfial model with parameter >.2.
The MLE's
for >. 1 and >.2 are >.1 and >.2, respectively. Find the value
J\
J\
of exp(A1-A2)8-11. Of ten lightbulbs tested, two fail before time t = 3. The failure time
distribution is assumed to be exponential. Determine the MLE of>..
Estimation of Parametric Survival Models
218
8.3 Univariate Models, Incomplete Data
8-12. Show that (8.25), (8.26) and (8.28) are equivalent.
8-13. Show that the likelihood given by (8.28) can also be written as
.
8-14. An exponential distribution with mean 50 years in truncated from
above at an unknown age z. A sample of five lifetimes from the trun­
cated distribution is 40.2, 55.7, 55.7, 60.5 and 72.1. From this data,
find the MLE of z.
8-15. A population is believed to be subject to a uniform survival distri­
bution with parameter w. A sample of five lives was observed from
time 5 to time 10. One death occurred at time 8, another death oc­
curred at time 9, and the other three lives survived to time l 0. Find the
MLE ofw.
8-16. From Equation (8.32), derive the equations which produce ci and [3.
A
8-17. Derive the equations for ci and {3 which minimize (8.35).
A
8-18. Using the data of Example 8.12, estimate the Weibull parameters k
and n by unweighted least squares.
8-19. Derive the three equations in dA , de and de by differentiating (8.39).
8-20. A Gompertz distribution is to be fit to a set of observed µi. The
parameter B = .000 I has been specified, and the parameter f3 = In c
A
is to be estimated by least squares. Give a formula for [3.
8.4 Hypothesis Testing of Parametric Models
A
8.21. From the S(t) determined in Example 8.8, calculate the value of the
x2 statistic. At what significance level could we reject the hypothesis
that S(t) is uniform?
Exercises
219
A
8.22. Repeat Exercise 8-21 for the S(t) determined in Example 8.14.
8.23. A sample of n persons is observed from exact age x until all die, and
deaths are grouped into quinquennial age groupings. sd:x+t denotes the
observed deaths in (x+t, x+t+5]. The last person died at age x+44.
An exponential distribution is to be fit to the data, with the parameter
determined as the minimum modified x2 estimator. Give an expres­
sion for the function to be minimized by this estimator.
8-24. In the following table, the values of ,lsqx are calculated from a fully
specified survival model, and the values of sdx+t are the observed
deaths from a complete mortality experience of 100 asthma sufferers,
age x at entry to the study.
t
0
5
10
15
20
25
30
35
40
,lsqx
.05
.08
.10
.15
.16
.16
.15
.10
.05
sdx+t
4
6
11
18
24
20
12
4
1
We hypothesize that the mortality of asthma sufferers is governed by
the specified model. By reference to a table of x2 probabilities, with
what level of confidence could we reject this hypothesis?
8-25. Suppose the values of ,lsqx in Exercise 8-24 are given by a Weibull
distribution with parameters estimated from the data. With what level
of confidence could we now reject the hypothesis?
8-26. From a group of five cancer patients diagnosed at t = 0, three died in
(0,1] and two died in (1,2}. We wish to fit the survival distribution
S(t) = k', t � 0, 0 < k < I, to this data. Assuming there exists a
minimum modified x2 estimator of k between O and l, it satisfies the
equation 30k3 +ak2 +bk+c = 0. Find a, band c.
Estimation of Parametric Survival Models
220
8-27. A sample of two lives at t = 0 experiences times of death t 1 = 5 and
t2 = 9. Let S(t) = 1 - /0 be the hypothesized uniform distribution.
Calculate A 2 by (8.48), and verify the result by using (8.49).
8-28. Calculate the modified A 2 statistic, from (8.50), for the hypothesized
A
exponential S(t) with .X estimated from the data of Example 8.2 by the
method of medians.
8-29. The following table gives approximate values of Pr (Y > y) for certain
critical values of y, the Kolmogorov-Smirnov statistic.
Pr(Y> y)
y
1.2238
.100
1.3581
.050
1.4802
.025
.010
1.6276
.005
1.7308
l .9495
.001
For a sample of n lives, a certain survival model has been hypothe­
sized. A complete set of times at death were observed from the
sample, and the Kolmogorov-Smirnov statistic was calculated. It was
found that the hypothesis was rejected at the 5% significance level,
but not at the 2.5% level. If the departure measure Dn were increased
by .005, then the hypothesis would be rejected at the 1% level, but not
at the .5% level. Find the largest possible integral value of n implied
by these facts.
8.5 Concomitant Variables in Parametric Models
8-30. Derive Equations (8.55).
8-31. Solve the three equations in Example 8.19 for t:i0 , fi 1 and t:i2 •
Exercises
221
8-32. For a linear-exponential survival model, the hazard rate function for
person i is of the form >.(t;z;) = a1z1; + a2z2;, where a1 and a2 are
parameters. All persons enter a study at time t = 0. We are given the
following table of data concerning the four persons of a study sample.
Person Time of
Departure
i
I
I
2
2
3
3
4
5
Method of Concomitant Variables
Departure
Zli
Z2;
Death
2
I
I
Death
I
Termination
4
4
8
8
Termination
Find the MLE's of a1 and a2.
8-33. A proportional hazard rate model is one in which >.(t;z;) and >.(t;z1),
the hazard rates for persons i andj, have a constant ratio regardless of
the value oft. If the underlying hazard >.(t) is not a constant function
oft, show that the Cox model (8.78) is a proportional hazard model,
whereas model (8.71) is not.
8-34. The Gompertz model >.(t) = Be' can be modified to include s con­
comitant variables by replacing B with B · exp
(t
J=I
a1z1 ).
(a) If the model is expressed as >.(t;z) = eta+a'z, what are k, ao and
z0, in terms of B and c?
(b) Show whether or not this new Gompertz model is a proportional
hazard model.
PART III
APPLICATIONS AND EXTENSIONS
In this final part of the text we will explore some extensions of the theory of
survival models and their estimation from sample data, as presented in the
first two parts of the text, as well as some interesting applications of that
theory.
Actuarial techniques, such as survival model analysis, are applicable in
many areas beyond those traditionally viewed as part of actuarial science. In
the next four chapters we will discuss such applications in the fields of
economics, finance, reliability engineering, epidemiology, and the natural
sciences, as well as traditional applications in actuarial science itself.
CHAPTER9
TRADITIONAL ACTUARIAL APPLICATIONS
9.1 INTRODUCTION
In Chapter 5 we briefly described how basic data about persons in a study
sample could be processed to obtain the ordered pair (y;,z;), the ages (or
times) at which person i entered and was scheduled to exit observation.
Chapters 6 and 7 were then devoted to the problem of estimating qx , the
conditional probability of death over (x, x+ I], given alive at age x, and in
Chapter 8 we considered the estimation of the parameters in a functional
form of S(x) or ,\(x). In both cases we saw how the ordered pairs (yi ,z;) were
used to obtain the desired estimates.
Before the advent of the computer age, the use of exact ages into and
out of a study for all persons in a large-sample study involved a prohibitive
amount of hand-calculation. As a result, grouping methods were developed
which reduced the level of computation without sacrificing much accuracy.
In this text, however, we make the reasonable assumption that all large-scale
studies are completely computerized, which allows us to take the exact age
approach to estimation. Furthermore, the exercises in this chapter anticipate
that the reader has access to computing facilities .
In this chapter we will first expand on the small-sample exact age il­
lustrations of earlier chapters by describing the data processing steps in­
volved in a large-sample study in which each person's actual date of birth is
utilized. Then in Sections 9.3 and 9.4 we will introduce the important con­
cepts of insuring ages andfiscal ages, and other aspectsiof study design that
are part of large-scale actuarial studies using insurance company or pension
fund data.
9.2 ACTUAL AGES
In Examples 5.1 and 5.2 we saw how the exact age y; at which person i came
under observation could be obtained from the date of birth and the latter of
the date on which the observation period opened and the date person i joined
the study sample. Now let us see how to streamline such calculations.
226
Traditional Actuarial Applications
9.2.1 Decimal Years
A convenient way to obtain exact ages is to first express dates of events in
tenns of decimal years. These, in turn, are easily obtained from a day-of-the-
year table, illustrated by Table 9.1, which has been stored in the computer.
TABLE9.1
Date Jan Feb
Mar
Apr May Jun Jul Aug
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91 121
92 122
93 123
94 124
95 125
96 126
97 127
98 128
99 129
100 130
101 131
102 132
103 133
104 134
105 135
106 136
107 137
108 138
1.09 139
110 140
111 141
112 142
113 143
114 144
115 145
116 146
117 147
118 148
119 149
120 150
151
l
2
....
4
5
6
7
8
9
10
11
12
13
14
15
16
11
18
19
20
21
?2
23
24
25
26
21
28
29
30
31
.)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
'"!')
,,.__
23
24
25
26
27
28
29
30
31
Sep Oct Nov Dec
152 182 213 244 274
153 183 214 245 275
154 184 215 246 276
155 185 216 247 277
156 186 217 248 278
157 187 218 249 279
158 188 219 250 280
159 189 220 251 281
160 190 221 252 282
161 191 222 253 283
162 192 223 254 284
163 193 224 255 285
164 194 225 256 286
165 195 226 257 287
166 196 227 258 288
167 197 228 259 289
168 198 229 260 290
169 199 230 261 291
170 200 231 262 292
171 201 232 263 293
172 202 233 264 294
173 203 234 265 295
174 204 235 266 296
175 205 236 267 297
176 206 237 268 298
177 207 238 269 299
178 208 239 270 300
179 209 240 271 301
180 210 241 272 302
181 211 242 273 303
212 243
304
305 335
306 336
307 337
308 338
309 339
310 340
311 341
312 342
313 343
314 344
315 345
316 346
317 347
318 348
319 349
320 350
321 351
322 352
323 353
324 354
325 355
326 356
327 357
328 358
329 359
330 360
331 361
332 362
333 363
334 364
365
Actual Ages
227
Sufficient accuracy can be obtained by ignoring leap years. Any events oc­
curing on February 29 will be presumed to have occurred on February 28.
The decimal part of a decimal year is then obtained by dividing the
day of the year by 365.
[i,XAMPLE 9.111 Find the decimal year for October 11, 1953.
�OLUTION! October 11 is day 284 of the year, so the decimal year would
be 1953 + 284/365 = 1953.78. (In the examples and exercises for this chap­
ter, decimal years will be rounded to two places.)
0
9.2.2 Exact Ages
It is easily seen that the exact age at which an event takes place is found by
subtracting the decimal year of birth from the decimal year of the event.
�XAMPLE 9.2!j If person i was born on October 11, 1953, and comes un­
der observation on August l, 1992, find Yi, the exact age that observation
begins.
!SOLUTION! August l, 1992 is decimal year 1992+213/365 = 1992.58.
0
Theny; = 1992.58-1953.78 = 38.80.
Example 9.2 illustrates the general approach to finding the exact age
at which an event occurs. For each person in the sample, three ages are
important: Y;, the age into a study; z;, the scheduled age out of the study; and
the actual age out of the study, due to death or withdrawal before age z;, if
this does, in fact, occur. We denote the exact age at death by O; and the
exact age at withdrawal by </>;, and adopt the convention that O; = 0 if person
i is not a study death, and </>; = 0 if person i is not a study withdrawal.
Therefore each person in a study is assigned an age vector defined
by v; = [y;, z;, O;, ¢;] which contains all the information needed to process
person i's contribution to the estimators presented in previous chapters.
!!EXAMPLE 9.3!1 An observation period runs from August l, 1992 to De­
cember 31, 1997. Person i was born on October 11, 1953, and dies on April
12, 1996. Find the age vector v; for this person.
[SOLUTION! The age into the study is y; = 38.80, from Example 9.2.
December 31, 1997 is decimal year 1997+365/365 = 1998.00, so the
228
Traditional Actuarial Applications
scheduled age out of the study is zi = 1998-1953.78 = 44.22. April 12
1996 is decimal year 1996+102/365 = 1996.28, so the age at death is found
to be B; = 1996.28-1953.78 = 42.50. The age at withdrawal is ¢; ::::: o
Thus vf = (38.80, 44.22, 42.50, O].
ci
To estimate qx over the interval (x, x+ I], it is necessary to first
determine person i's contribution, if any, to (x,x+l]. We note first that if
y; � x+ 1, then person i enters the study beyond age x+ I and does not
contribute to (x,x+1]. Similarly, if z; '.S x, then person i leaves the study
before age x and does not contribute to (x,x+1]. Even if y; < x+1 and
z; > x, which implies that person i is scheduled to contribute to (x,x+1], the
occurrence of death or withdrawal before age x would prevent this scheduled
contribution from being realized. This would be evidenced either by
0 < O; < x or by O < </>; < x.
Once the age vectors for persons who do not contribute to (x, x+ l]
have been eliminated from consideration, the next step is to convert each age
vector v; into a duration vector for the interval (x,x+l], showing the frac­
tional duration within (x, x+ 1] at which observation begins, is scheduled to
end, or actually ends due to death or withdrawal prior to scheduled ending.
We denote the duration vector by uf,x = [r;,s;, i;, K;], where, necessarily, at
least one of l; and K; must be zero. The double subscript on u' identifies
both the person and the estimation interval to which it relates.
The reader should verify the following conversion relationships for
person i known to contribute to (x, x+1].
=
r;
Si
l;
C-x
=
=
t-x
E-x
ify; '.S x
if x < Yi < x+l
ifx < Z; < x+l
if Z; � x+l
if O; = 0
ifx < B; '.S x+ 1
ifB; > x+l
(9.la)
(9.lb)
(9. lc)
Actual Ages
229
K,;
G·-x
if <Pi = 0
ifx<ip; -:; x+ 1
(9.ld)
if¢; > x+l
!!EXAMPLE 9.4ij Convert the age vector of Example 9.3 into duration vec­
tors for the estimation intervals (38,39], (39,40], and (42,43].
!SOLUTION! For (38, 39], we have r; = y;-38 = .80, s; = 1 (since
z; > 39), l; = 0 (since O; > 39), and "'; = 0 (since cp; = O); thus
u!,3s = [.80, 1, 0, O]. For (39,40], r; = 0 (since y; < 39), s; = I (since
z; > 40), l; = 0 (since B; > 40), and "'; = 0 (since </>; = O); thus u!,39 =
[O, 1, 0, OJ. For the interval ( 42, 43], r; = 0 (since y; < 42), s; = 1 (since
z; > 43), l; = B;-42 = .50, and"'; = 0 (since ¢;=0); thus uf,42 = [O, 1, .50, OJ.
u!,39 = u!,40 = uf.4 1, and uf.x is not defined for x < 38 or for x > 42, since
0
person i does not contribute to any such intervals.
9.2.3 Calculation ofExposure
Many of the estimators developed in Chapters 6 and 7 are of the general
form fix = /;;:!'::re. This includes the moment estimators (which use sched­
uled exposure), the actuarial estimators (which use actuarial exposure), and
the full data, exponential assumption MLE's (which use exact exposure).
Estimators of this form are particularly easy to apply when the basic
data is represented by u;,x for person i within interval (x, x+ 1]. We first note
that the number of observed deaths in (x, x+ 1] is simply the number of u;,x
vectors for which l; f. 0. Similarly the number of observed withdrawals in
(x,x+ IJ is the number of U;,x vectors for which"'; f. 0.
The exact exposure over (x, x+ I] contributed by person i is simply
(Exact Exposure),, � [ � ] - r, ,
where
[f]
(9.2)
represents the minimum of s;, t;, "'; that exceed zero. In other
words, if t; = "'; = 0, so that person i neither dies nor withdraws in (x, x+ I],
then we have (Exact Exposure);,x = s;-r;. But if person i dies in (x, x+ l J,
so that l; < s; and "'; = 0, then (Exact Exposure);,x = t;-r;. FinaJJy, if
230
Traditional Actuarial Applicatio113
person i withdraws in (x,x+ l], so that K; < s; and l; = 0, then we have
(Exact Exposure);,x = K,;-r;.
To find the scheduled exposure for estimating the mortality proba­
bility q: (d) under Hoem' s moment approach, exact exposure is still used for
those who withdraw and for those who neither die nor withdraw in (x, x+ l],
but those who die are exposed to age x+s;. Thus we have
(Scheduled Exposure);,x =
[ �;] - r;,
(9.3)
where ,.,,; is used if person i withdraws in (x, x+ l], and s; is used otherwise.
The actuarial exposure differs from the other two types by the
convention of exposing deaths in (x, x+ l] to age x+ 1. Thus we have
(Actuarial Exposure),., =
[S;]
�·
- r1,
(9.4)
where ""; is used if person i withdraws in (x, x+ 1], S; is used if person i does
not die or withdraw in (x,x+ 1 ], and 1 is used if person i dies in (x, x+ 1 ].
jlEXAMPLE 9.5!1 Consider the age vector vf = [39.85,40.75,40.25,0].
Find each of (a) (Exact Exposure);,4o, (b) (Scheduled Exposure)1,40, and (c)
(Actuarial Exposure),,40·
[SOLUTION! First we convert vf to u(,40 = [O, .75, .25, OJ. Then we find
(a) (Exact Exposure);,40 = .25, (l; < s1);
(b) (Scheduled Exposure);,40 = .75, where we use s;-r;, rather than l;-r;;
0
(c) (Actuarial Exposure)1,40 = I, where we use 1-r; rather than l;-r;.
The total exposure over (x, x+ I] is then found by summing over all persons
who contribute to (x,x+ I]. Thus
(Exposure)x =
Li ( Exposure);,x,
(9.5)
where (9.5) holds for all three types of exposure. The reader should recog­
nize that the three types of exposure differ from each other only with respect
to the treatment accorded to deaths within (x,x+l].
Actual Ages
231
9.2.4 Grouping
In many cases an average age at event might be substituted for the exact age
at event for all persons whose exact age at event falls within a certain age
range. For example, consider all persons whose exact y/s fall between the
integers x and x+ 1. We might substitute a common age y' for all such y;,
frequently using y' = x+ f as the assumed average value of they; 's. When
we do this the entrants to the study have been grouped by age last birthday,
since it is those with a common age last birthday (x) that are being con­
sidered together.
A second type of grouping is one that is done by calendar age. Cal­
endar age at event is defined to be the integral age y obtained on the birthday
in the same calendar year in which the event takes place. For example, if a
person's date of birth is September 14, 1960, and date of withdrawal is June
26, 1994, then the calendar age at withdrawal is 34, since this person would
be integral age 34 on the birthday in the calendar year of withdrawal. Calen­
dar ages at event are easily found by
CA = CYE - CYB,
(9.6)
where CYE is the calendar year of the event and CYB is the calendar year of
birth.
It is easy to see that a person with calendar age w at event has an
exact age at event that falls within (w-1, w+ 1). If the w'h birthday is
January 1, and the event is December 31 of the same year, then the calendar
age at event is w but the exact age is practically w+ 1. Conversely, if the w1h
birthday is December 31, and the event is January 1 of the same year, then
the calendar age at event is still w, but the exact age is virtually w-1.
The advantage of a calendar age grouping is that an integral age w
can then be reasonably substituted for the various exact fractional ages.
When withdrawals are grouped by calendar age, and assumed to occur at the
integral age w, then all withdrawals are occurring at the boundaries of the
various estimation intervals. The occurrence of withdrawals only at estima­
tion interval boundaries greatly simplifies the estimation work, since it cre­
ates a Special Case A estimation problem.
!!EXAMPLE 9.61! An observation period begins on January I, 1990. Con­
sider a person born on April 12, 1970 who joins the group under observation
on February 2, 1991. What is this person's exact age at entry to the study?
What age would be substituted for the exact age if calendar age grouping is
used? If age last birthday grouping is used?
Traditional Actuarial Applications
232
!SOLUTION! Frnm Table 9.1, April 12, 1970 is 1970.28 and February 2,
1991 is 1991.09. Then the exact age is y; = 1991.09-1970.28 = 20.81. The
calendar age is CA = 1991-1970 = 21. The age last birthday is 20, so the
substituted age would be 204.
D
9.2.5 Actual Age Summary
Here we recapitulate the steps involved in processing basic data for the
purpose of estimating qx (or a constant force of mortality) over the estima­
tion interval (x, x+ l ]. We assume that the basic data for each person
consists of the date of birth, date of joining the group under study (e.g., date
of insurance policy issue), and date of death or withdrawal, if applicable. In
addition, the opening and closing dates of the observation period (O.P.) must
be specified. Actual ages are used.
(l) Any person whose death or withdrawal occurs before the O.P.
opens, or whose date of joining the group occurs after the O.P.
closes, is not part of the study sample.
(2) Dates of birth, joining the group, and death or withdrawal are
converted to decimal years.
(3) Age at entry to study, y;, is found as age when 0.P. opens, or
age when person i joins group, whichever is later.
(4) Scheduled age at exit from study, z;, is found as age on closing
date of O.P.
(5) Age at death or withdrawal, B; or <Pi, is found from date of event
and date of birth; steps (3), (4) and (5) produce the age vector
vf = [y;, z;, B;, <p;].
(6) The duration vector u!,x = [r;, S;, L; t.:;] is obtained from vf with
respect to the estimation interval (x, x+1].
(7) The estimators presented in Chapters 6, 7 and 8 can then be
evaluated from the u;,x vectors for all persons in the study.
In Section 9.2.3 we discussed the evaluation of all estimators of the general
form /;;;;'::,e. Estimators for Special Case C, such as (6.19), which make
use of ex , the number scheduled to end observation in (x, x+1], can also be
easily evaluated from the u;,x vectors. Recall that Special Case C has r; = 0
for all i. Then all vectors which shows; < l belong to ex for the estimation
interval (x,x+ l].
Actual Ages
233
The V; vectors also contain all the information needed to evaluate the
various Chapter 8 estimators for the parameters of a hypothesized parametric
survival model. In this case it is not necessary to convert v; into u;,x. As
seen in Chapter 8, it will frequently be necessary to evaluate these estimators
by numerical methods.
The exercises presented at the end of this chapter give the reader
ample opportunity to become comfortable with the required data processing
steps. As already mentioned, it is assumed that the reader has access to
adequate computing facilities.
9.3 INSURING AGES
Since most people do not purchase individual insurance policies on their
actual birthdays, it follows that they are various fractional attained ages
when their policies are issued. The insurance company, on the other hand,
will not calculate a premium for the policy that depends on the insured's
actual fractional age at issue. Instead an integral insuring age is substituted
for the insured's actual age as of the policy issue date. Most commonly this
insuring age will be the insured's actual age on the birthday nearest the
policy issue date. In other words,
IA = Actual Age Nearest Birthday,
(9.7)
where IA is the insuring age.
Less commonly, the insuring age could be taken as the actual age on
the birthday preceding the policy issue date, in which case we say that IA is
the actual age last birthday. Note that the examples and exercises in this
chapter all use the age nearest birthday basis.
9.3.1 Valuation Year of Birth
Assigning an integral insuring age to the insured as of the policy issue date
implies that a hypothetical date of birth, called the insuring date of birth,
has been substituted for the actual date of birth. Clearly the month and day
of this insuring date of birth are the same as the policy issue date. The
hypothetical year of birth, called the valuation year of birth (VYB), is then
found as
VYB = CY! - IA,
(9.8)
234
Traditional Actuarial Applications
where CY! is the calendar year of policy issue. It should be recognized that
VYB will frequently be the same as CYB, but it can also be one year earlier
or one year later than CYB.
IIEXAMPLE 9.7lj An insurance policy is issued on August 22, 1998. Find
the IA and the VYB if the actual date of birth is (a) January 12, 1978; (b) July
4, 1977.
!SOLUTION!
(a) As of August 22, 1998 the nearest birthday is the one coming up on
January 12, 1999, when this person will be IA= 21. Then we have
VYB = 1998-21 = 1977, which is one year less than the actual CYB.
(b) As of August 22, 1998 the nearest birthday is the one just passed on
July 4, 1998, so that IA = 21. Then VYB = 1998-21 = 1977, the same
as the actual CYB.
D
9.3.2 Anniversary-to-Anniversary Studies
When insuring ages are used, a natural choice for the observation period is
one that opens on the policy anniversary in a designated year for each
insured person involved in the study. Similarly, the observation period would
close on the policy anniversary in a later year. For example, an observation
period might be defined as running from policy anniversaries in 1994 to
those in 1998. Note that each person involved in the study has his or her
own observation period.
Observation periods that· run from a fixed date to a later fixed date
can be used with insuring ages, but there are definite advantages to the anni­
versary-to-anniversary study when insuring ages are used. In this text we
emphasize anniversary-to-anniversary studies with insuring ages.
The major convenient consequence of anniversary-to-anniversary
studies with insuring ages is that all persons enter the study at an integral age
Yi· This is true whether the person enters the study by joining the group via
policy issue during the O.P. (at an integral insuring age), or by already being
in the group when the O.P. opens. In the latter case, entry is at a policy anni­
versary which is always the attainment of an integral (insuring) age. Since Yi
is an integer, it follows that ri = 0 for any estimation interval (x,x+ l], since
x is integral.
Similarly, with the O.P. ending on a policy anniversary, all sched­
uled ending ages z; are integers, from which it follows that s; = 1 for all
estimation intervals (x,x+1]. Hence all vectors u; x are of the convenient
Insuring Ages
235
form [O, 1, l;, K:;], and all anniversary-to-anniversary insuring age studies
belong to our Special Case A.
If there are no withdrawals, then the estimate is found from (6.7)
simply by counting the number of vectors U;,x with r; = 0, which gives nx ,
and the number with l; #- 0, which gives dx, for (x, x+ I]. If there are with­
drawals, the number of them in (x,x+ I], wx , is the number of vectors u;,x
with K:; =I= 0. With nx, dx and Wx available, q:<d) and q: (w) can be estimated
by (6.34) or (6.37). Of course any of the exposure-based estimators could be
used as well. Note that since all s; = 1, the moment estimator and the ac­
tuarial estimator are the same.
Another approach that is frequently followed when withdrawals are
present in an anniversary-to-anniversary insuring age study is to group the
withdrawals by calendar insuring age. This is similar to actual calendar
age, as defined by (9.6), exceptthat VYB is used instead of CYB. Thus
t
CIA = CYW - VYB,
(9.9)
where CIA stands for calendar insuring age, and CYW is the calendar year of
withdrawal. Then all withdrawals with CIA = w, an integer, are assigned
</); = w in place of their actual </J; ages. Since all withdrawals are then
viewed as occurring at an integral insuring age, which is an estimation
interval boundary, so that none occur within (x, x+ 1 ], then the simplified
Special Case A again results, and qx can be easily estimated by (6.7).
It should be noted that grouping by CIA is the same as assuming that
all withdrawals occur on policy anniversaries. Since withdrawal from an
insured group generally means policy surrender, which frequently occurs on
a policy anniversary anyway, the assumption of all withdrawals on policy
anniversaries is not unreasonable.
!!EXAMPLE 9.Slj For the sample of ten policyholders shown below, esti­
mate q30 under the following conditions:
( l ) The observation period runs from anniversaries in 1993 to
those in 1998.
(2) Insuring ages are used throughout.
(3) Withdrawals are grouped by CIA.
Traditional Actuarial Applications
236
Person
1
2
3
4
5
6
7
8
9
10
Date of
Birth
Mar 17, 1964
May 6, 1964
Aug 12, 1964
Oct 27, 1964
Jan 4, 1965
Apr 18, 1965
May 20, 1965
Jul 4, 1965
Sep 16, 1965
Dec 11, 1965
Date of
Policy Issue
Jun 20, 1992
Aug 6, 1992
Dec 18, 1992
Jan 4, 1993
Apr 28, 1993
Jun 16, 1993
Oct 29, 1993
Feb 16, 1994
Aug 22, 1994
Mar 6, 1995
Date of
Death
Date of
Withdrawal
-
-
Jun 12, 1993
-
-
Jun 18, 1995
-
Dec 12, 1995
-
Feb 22, 1997
-
Aug 29, 1996
-
-
Apr 21, 1996
-
Feb 17, 1997
ISOLUTIONI Each person is first assigned an IA by (9.7) and a VYB by
(9.8). Then the VYB, Date of Policy Issue, and dates of the O.P. imply an
integral y; and z; for all persons involved in the study. The grouping assump­
tion for withdrawals implies that an integral </>; replaces the exact, fractional
</>;. Exact ages at death, B;, are fractional. The vector v; then implies the vec­
tor U;,30· The data are summarized in the following table.
Person
1
2
3
4
5
6
7
8
9
10
IA
28
28
28
28
28
28
28
29
29
29
VYB
1964
1964
1964
1965
1965
1965
1965
1965
1965
1966
Age Vector v;
Z;
B;
<Pi
- -
-
-
y;
29
29
28
28
28
28
29
29
29
34
34
33
33
33
33
33
33
32
0
0
0
31.34
0
30.48
0
0
30.95
0
31
0
0
30
0
0
32
0
Duration Vector u;,J(j
r;
i;
S;
K-;
o 1 0 0
0
0
0
0
0
0
0
-
-
-
-
·- ·-·
l
1
l
l
.48
0
0
.95
0
0
0
0
-
l
1
1
0
0
0
1
0
0
Note that Persons 2 and 6 do not contribute to (30,31). Person 2 is not even
in the study, since death precedes the O.P., and Person 6 withdraws at
CIA = 30, so does not enter (30,31). The remaining n30 = 8 persons all
enter (30, 31) at age 30 (i.e., r; = 0), and stay until age 31 (i.e., s; = 1 and
r;,1 /. s;), except for the d30 = 2 deaths (Persons 7 and I 0). As expected, this
study design produces a Special Case A estimation situation. Using Estimator (6.7), we find q30 = = .25.
D
j
237
Insuring Ages
[ltXAMPLE 9.911 Repeat Example 9.8, calculating the full data MLE of
q30, assuming a constant force of mortality over (30, 31 ).
�OLUTIONI This time we calculate exact exposure as I year for all 6
persons who do not die in (30, 31], .48 for Person 7, and .95 for Person 10.
A
2
dA
-2'74'
0
Thenµ= 7_43, an q30 = I-e ' · ., = .23599.
9.3.3 Select Studies
for the purpose of estimating S(t;x), as defined in Section 1.2, we consider
only those policies issued at IA = x. Again assuming an anniversary-to­
anniversary observation period, the data processing is quite similar to that
described for insuring age studies, with the following parallel features:
(1) The vector v; now represents the policy durations at entiy,
scheduled exit, death or withdrawal, rather than the insuring
ages of the insured at such events. Note that for policies issued
during the O.P., the duration at entiy to the study is 0.
(2) The vector u;,1 represents the location of these events within
estimation interval (t, t+ I].
(3) Withdrawals are grouped by calendar duration, instead of cal­
endar insuring age. Note that calendar duration is defined by
CD = CYW - CYI.
(9.10)
An example will show the similarity of select studies to insuring age studies.
�EXAMPLE 9.1011 Consider the policies of Example 9.8 with IA= 28. Es­
timate ql281+2 using an observation period that runs from anniversaries in
1994 to those in 1999, grouping withdrawals by calendar duration.
!SOLUTION! Again we expect Special Case A as a result of the study
design. In Example 9.8, Policies l through 7 have IA = 28, and Policy 2 is
not involved in the study due to death before the O.P. opens. The other poli­
cies have duration vectors for the study, V;, as follows:
Traditional Actuarial Applications
238
Duration Vector v;
Policy Yi z;G' B; </>;
2
0
I
7
0
2
3
3
7
0
4
6
1
0
0
3.34 0
5
l
6
6
2
1
0
6
7
l
6
2.48 0
For estimation interval (2,3], all policies enter at t =2 (i.e., ri= O), except
Policy 6 which does not contribute to (2,3] due to the calendar duration
grouping of withdrawals. All policies which enter are scheduled to complete
(2,3], and all do except Policy 7. Using any of the moment estimator, the
partial data MLE, or the full data MLE with uniform distribution assump­
tion, we have qr2si+2 = = .20000. Using the full data MLE with exponen­
tial distribution assumption, we have ql281+2 = l -e-1/4.4s = .20005.
D
!
9.3.4 Insuring Age Summary
In this section we have shown how the combination of insuring ages with an
anniversary-to-anniversary observation period, the insuring age analogy to
an actual age birthday-to-birthday study, considerably simplifies the esti­
mation ofqx over the interval (x, x+ 1]. Note that basic data was available on
each insured life in the sample.
If a date-to-date observation period is used, then ages at entry to and
exit from the study, y; and Zi, will no longer be integral. This situation is the
same as that presented in Section 9.2 using actual ages.
Instead of grouping withdrawals by calendar insuring age, we might
group them by age last birthday. That is, all withdrawals occurring in
(x,x+l], i.e., at age x last birthday, are assumed to occur at some average
age, commonly .x+
In this text we take the point of view that an anniversary-to­
anniversary observation period, with withdrawals grouped by calendar insur­
ing age, are the features of a standard insuring age study using data from
a sample of individual life insurance policy-holders. Date-to-date studies
and/or withdrawals grouped by age last birthday constitute variations of the
standard study design.
f.
Fiscal Ages
239
9.4 FISCAL AGES
In the case of group insurance or group pension plans, a large number of
individual persons are covered under a single policy or plan. There will exist
a key date for such a plan, called the plan anniversary or plan valuation
date. On such a date it will be necessary to calculate premium rates, actu­
arial present values of accrued benefits, or other financial values. For this
purpose it will be convenient for all members of the plan to be an integral
age on this key date. This integral age is called thefiscal age.
Note that this situation is similar to that under insuring ages, where
each individual insured was an integral insuring age on that person's policy
anniversary. Here the same idea holds, with the further condition that the
policy anniversary is the same_date for all persons. It is traditional to refer to
this date as the T-date.
Historically the terms T-date and fiscal age were adopted to indicate
that the T-date was the terminal date of the fiscal year of an enterprise. In
this text we feel that the major application of the fiscal age concept is to
studies ofmortality under group insurance or pension plans. Thus the T-date
is the plan anniversary. The term fiscal age is not particularly descriptive,
but we will retain it for the sake oftradition.
9.4.1 Fiscal Year ofBirth
Analogous to the definitions of insuring age and valuation year of birth in
Section 9.3, we assign each person in a group plan a fiscal age (FA) as of
some particular T-date, say the T-date in calendar year z. This fiscal age
would likely be the actual age nearest birthday on that date, or it could be the
actual age last birthday, or the actual calendar age. Regardless of how it is
assigned, we then define the fiscal year ofbirth (FYB) as
FYB = z-FA.
(9.11)
Just as was true for VYB under insuring ages, FYB could be the same as the
person's actual CYB, or it could be one year less or one year greater. Once
FYB has been assigned, the T-date in the FYB is then the hypothetical date of
birth for each person in the group plan.
IIEXAMPLE 9.1 tlj A pension plan has its anniversary on March 31. A new
employee becomes a member of the plan on March 31, I 998. This employ­
ee's actual date ofbirth is November 17, 1966. Ifthe fiscal age is assigned as
actual age last birthday, find this employee's FYB.
Traditional Actuarial Applications
240
!SOLUTION! The actual age last birthday as of March 31, 1998, is 3 J
which was attained on November 17, 1997. Then FYB = 1998-31 = 196/
Note that FYB = CYB+ l.
0
9.4.2 Observation Periodsfor Fiscal Age Studies
The natural choice of an observation period is one that runs from the I-date
in a certain year to the T-date in a later year. Note that the T-date is the anni­
versary for all members of the group plan, so a T-to-T study is both a date­
to-date study and an anniversary-to-anniversary study.
The principal benefit of using a T-to-T observation period is that all
members in the plan when the O.P. opens enter the study at an integral age
y;. Similarly, all members in the study sample will have an integral sched­
uled exit age z;. In turn, we know that integral y; and z; imply r; = O and
s; = l for any estimation interval (x,x+ l], and a Special Case A estimation
problem.
Although dates other than I-dates can be used in date-to-date fiscal
age studies, there is no particular advantage to this, and only T-to-T studies
will be considered in this chapter.
9.4.3 New Members and Withdrawals
lf new employees can only join the group plan under study on a T-date, then
such new employees would enter the study at integral fiscal ages, just as was
true under insuring ages. However such a restriction does not usually hold,
so persons joining the group during the observation period could do so at any
fractional age. Similarly, persons might withdraw from the group plan by ter­
minating employment at any date, and hence at any fractional fiscal age.
Recall that y; is the age at which person i enters the study, whether
as a member when the O.P. opens (in which case Yi is an integer), or as a
new member during the O.P. (in which case y; can be fractional).
The procedures using exact actual ages described in Section 9 .2 can
be used with exact fiscal ages as well. On the other hand, grouping proce­
dures could be used for the new members and withdrawals. If calendar
fiscal age grouping is used, then an integral fiscal age is substituted for the
exact fractional fiscal ages. If age last fiscal birthday grouping is used, then
usually y+ is substituted for all exact fractional ages in (y, y+ 1]. Whether
calendar age or age last birthday is used, it is customary to use the same
grouping for both new members and withdrawals.
!
241
Fiscal Ages
l[l:XAMPLE 9.121! A group life insurance plan has a June 30 anniversary
date. From the following sample data, estimate q40 using an O.P. that runs
from June 30, 1990 to June 30, 1995. Group new members and withdrawals
by fiscal age last birthday, assume an average age of y+ and use the actu­
arial estimator.
!,
Member
1
2
3
4
5
6
7
8
9
10
FYB
1952
1952
1951
1951
1951
1950
1951
1951
1953
1953
Date of Plan
Membership Date of Birth
May 12, 1989
Aug 24, 1989
Oct 3, 1989 Mar 17, 1994
Jan 30, 1990
May 18, 1990
Jan 3, 1991
Jul 15, 1990
Apr 7, 1992 Jun 22, 1992
Sep 15, 1994
Jul 1, 1995
Date of
Withdrawal
Sep 30, 1993
-
Apr 30, 1990
-
-
Feb 15, 1992
-
-
!SOLUTION! Note that Member 4 withdrew from the plan before the O.P.
opened, and Member 10 did not join the plan until after the O.P. closed, so
neither member is involved in this study. Members 6 through 9 are new
members during the O.P., whereas the others all come under observation on
June 30, 1990. The table on the following page gives y;, the fiscal age into
the study (exact age for starters, grouped age for new members); zi, the
scheduled fiscal age at exit from the study; 9;, the fiscal age last birthday at
death; ¢,;, the grouped fiscal age at withdrawal.
Member 9 does not contribute to (40, 41], since y; > 41. Members 6
and 8 enter (40, 41] at 40.5, and the 5 others enter at 40. Member 7 leaves at
40.5, and all others leave at 41, except Member 8 who dies in (40,41]. But
under the actuarial estimator, Member 8 is exposed to age 41 nonetheless.
Thus we have q40 =
- = 5\ = .18182.
s+i(\) !
242
Traditional Actuarial Applications
Member Yi
1
38
2
38
3
39
5
39
6
40.5
7
39.5
8
40.5
41.5
9
Z;
43
43
44
44
45
44
44
42
Oi
0
0
42
0
0
0
40
0
<Pi
41.5
0
0
0
0
40.5
0
0
0
9.4.4 Fiscal Age Summary
The use of fiscal ages gains an advantage over the use of insuring ages in
that all persons in the study sample attain integral ages on the same date.
Thus date-to-date and anniversary-to-anniversary studies become identical
whenever T-to-T is used for the O.P.
On the other hand, under insuring ages persons who join the group
under study during the 0.P. do so at an integral age, so only withdrawals can
occur at fractional ages. Under fiscal ages, both new members during the
O.P. and withdrawals can occur at fractional ages. When grouping is used, it
is customary to consider both categories together.
9.5 SUMMARY
With considerable attention having been paid to the theory of estimation in
prior chapters, our purpose in this chapter has been to describe the actuarial
practice of estimating qx over (x, x+ I], with emphasis given to insurance
company and pension plan situations. In particular the use of hypothetical
insuring ages (for individual policy studies) and fiscal ages (for group
studies) was introduced, and certain practical advantages were seen to follow
from this practice.
Considerable emphasis was given to the use of exact ages, with age
grouping considered as well. In the exercises which follow, both approaches
will be used. We repeat that most of the exercises assume that the reader has
access to adequate computing facilities.
Exercises
243
9.6 EXERCISES
The exercises for this chapter are quite different from those presented at the
end of previous chapters. It is our expectation that the twelve examples in
this chapter have provided the reader with an adequate reinforcement of the
relevant theory and procedures. Accordingly, the following exercises provide
the reader with an opportunity to deal with our topic in a macro, rather than
micro, sense.
On pages 244 and 245 the basic data are given for a sample of 70
persons, showing Date of Birth, Date of Entry, Date of Withdrawal, and
Date of Death. The sample data will be used in three different contexts in the
exercises, where each context requires a slightly different meaning for the
concept of Entry. This will be made clear in each of the three sets of exer­
cises that follow.
9.1 Introduction;
9.2 Actual Ages
For Exercises 9-1 through 9-4, the data refer to members of a fraternal or­
ganization. In particular, the Date of Entry is the date of membership in the
organization. Values of qx are to be estimated from this sample data using an
observation period of January 1, 1990 to December 31, 1999.
9-1. Determine the exact actual age vector v; for each member of the sam­
ple, indicating any members who are not involved in the study. Round
decimal years to two decimal places.
9-2. Find the values of dx , the number of observed deaths in (x, x+ 1 ], for
all values of x for which dx -=/:- 0.
9-3. For alt values of x for which exposure exists, use the results of Exer­
cise 9-1 to find each of the following:
(a) (Exact Exposure)x
(b) (Scheduled Exposure)x
(c) (Actuarial Exposure)x
9-4. For all values of x for which dx -=/:- 0, estimate qx by each of the fol­
lowing:
(a) Full data MLE, exponelltial distribution assumption.
(b) Grouped actuarial estimator, using age last birthday grouping for
study starters and enders, and calendar age grouping for new mem­
bers and withdrawals during the O.P.
244
Traditional Actuarial Applications
Person
Number Date of Birth
Date of Entry
Date of
Withdrawal
1
2
3
4
5
Apr 12, 1960
Jul 2, 1956
Jun 11, 1958
Feb 28, 1954
Nov 28, 1961
Jan 16, 1988
Mar 29, 1988
Aug 1, 1988
Aug20, 1988
Jan 11, 1989
6
7
8
9
10
Jun 6, 1951
Mar 15, 1957
Aug 24, 1958
Jun 25, 1960
Feb 23, 1959
Jan 6, 1989
Dec 9, 1988
May 3, 1989
Jan 31, 1990
Sep 22, 1989
May 3, 1989
11
12
13
14
15
Dec 15, 1957
Jul 6, 1958
Jun 10, 1960
Feb 19, 1959
Oct 15, 1960
Oct 8, 1989
Feb 18, 1990
Apr 5, 1990
Jun 2, 1990
Nov 24, 1990
May 16, 1991
Aug10, 1999
16
17
18
19
20
Apr 14, 1961 Feb 10, 1991
Aug10, 1958 Dec 20, 1990
Oct 8, 1961 Jul 7, 1991
Oct 26, 1959 May 26, 1991
Feb 5, 1960 Dec 16, 1991
21
22
23
24
25
Dec 14, 1961
Dec 10, 1960
Jul 31, 1962
Jan 24, 1961
Jul 18, 1963
Dec 12, 1991
May 2, 1992
Apr 22, 1992
Sep 9, 1992
Jan 28, 1993
26
27
28
29
30
Oct 2, 1962
Sep 5, 1960
Mar 21, 1961
Apr 10, 1963
May 25, 1962
Oct 14, 1992
Aug24, 1992
Nov 12, 1992
Mar 4, 1993
Dec 30, 1992
31
32
33
34
35
Aug 6, 1962 May 26, 1993
Nov 20, 1963 Jun 13, 1993
Dec 10, 1961 Apr 6, 1993
Aug 4, 1963 Sep 2, 1993
Sep 10, 1963 Feb 28, 1994
Dec 1, 1990
Aug27, 1989
Mar4, 1990
Dec 31, 1990
May 10, 1998
Apr 6, 1996
Date of Death
Jan 4, 1990
Dec 25, 1991
May 10, 1998
Apr 14, 1996
Oct 10, 1997
Aug25, 19-94
Mar 6, 1997
Sep 9, 1993
Nov 16, 1999
Jun 4, 1997
Nov 14, 1994
Dec 2, 1997
Jun 21, 1995
Aug19, 1994
Feb 15, 1998
245
Exercises
Person
Nu mber Date of Birth
Date of Entry
36
37
38
39
40
Jun 30, 1962
Feb 28, 1963
Jan 31, 1964
Sep 5, 1962
Nov 2, 1963
Nov l, 1993
Jul 15, 1994
Aug 8, 1994
Nov 2, 1994
Jan 21, 1995
41
42
43
44
45
Jul 15, 1965
Apr 20, 1964
Jul 27, 1965
Dec 3, 1965
Jan 4, 1964
Mar 16, 1995
May 4, 1995
Sep 17, 1995
Apr 10, 1996
Dec 23, 1995
46
47
48
49
50
Nov 6, 1964
May 19, 1966
Aug 16, 1966
Jan 4, 1965
Mar IO, 1966
Mar 10, 1996
Jul 12, 1996
Jul 6, 1996
Aug 19, 1996
Jun 25, 1996
Jun 21, 1996
Jul 12, 1997
Jul 6, 1999
51
52
53
54
55
Sep 24, 1966 Feb 7, 1997
Nov 15, 1966 Apr 14, 1997
Sep 6, 1964 Dec 8, 1996
Sep 14, 1967 May 21, 1997
Nov 21, 1965 Aug 27, 1997
Dec 21, 1998
56
57
58
60
Aug 19, 1967
Jan 6, 1967
Apr 6, 1967
Aug 4, 1965
Feb 3, 1966
Sep 16, 1997
Aug 9, 1997
Nov 30, 1997
Oct 23, 1997
Dec 4, 1997
61
62
63
64
65
Apr 27, 1968
Feb 26, 1967
Aug 13, 1968
Jul 6, 1966
May 21, 1969
Jan 4, 1998
May IO, 1998
Jul 25, 1998
Sep 16, 1998
Feb 3, 1999
66
67
68
69
70
Sep 15, 1966
Nov 8, 1967
Jul 21, 1969
Feb 16, 1968
Nov 13, 1966
Dec 12, 1998
Apr 27, 1999
Aug 5, 1999
Oct 18, 1999
Jan 14, 2000
59
Date of
Withdrawal
Mar 10, 1998
Date of Death
Jul 16, 1997
Aug 6, 1996
Jul 6, 1998
Jan 20, 1996
Sep 24, 1998
Sep 21, 1997
Nov 23, 1997
Jun 4, 1998
Aug 19, 1998
Jun 4, 1998
Nov 19, 1999
Dec 12, 1999
Jan 10, 2000
Sep 5, 1999
246
Traditional Actuarial Applications
9.3 Insuring Ages
For Exercises 9-5 through 9-7, the data refer to a sample of individual life in.,
surance policyholders, and the Date of Entry is the date of policy issue. In­
suring age is defined as actual age nearest birthday at issue. The observation
period will be policy anniversaries in 1990 to those in 1999. Group with­
drawals by calendar insuring age.
9-5. Find the insuring age vector v; for each policyholder, indicating those
not involved in the study. Use the calendar insuring age, rather than
exact insuring age, for </>;.
9-6. Recall that this study design produces a Special Case A study where
= ndx . Calculate dx and nx for all x for which nx i- 0.
x
t
9-7. For each of issue ages 30, 31 and 32 separately, a select study is to be
performed. Use the observation period of A, 1990 to A, 1999 and
group withdrawals by calendar duration, so that Special Case A again
results. Calculate d[x]+t and n[x]+t, for x = 30,31,32, for all t for which
nrxl+t i- 0.
9.4 Fiscal Ages
For Exercises 9-8 through 9-10, the data refer to the members of a group
insurance plan with policy anniversary on June 30. The Date of Entry is the
date of employment, at which time insurance coverage begins. A fiscal age
is assigned as the actual age nearest birthday as of the June 30 following
Date of Entry. (E.g., for Date of Entry November 18, 1995, the following
June 30 is June 30, 1996. If the member's actual birthday is April 4, 1964,
then the fiscal age is the actual age nearest birthday on June 30, 1996, which
is 32. Then FYB = 1996-32 = 1964.) Values of qx are to be estimated over
an observation period of June 30, 1992 to June 30, 2000.
9-8. Find the set of exact fiscal age vectors v; for all persons who are
involved in this study.
9-9. Using the exact fiscal age vectors, estimate qx by the full data, expo­
nentialassumption, MLE, for all x for which dx i- 0.
9-10. Calculate the. actuarial exposure for all intervals (x, x+ 1] for which the
exposure is non?ero, separately for each of the following grouping
assumptions for new members and withdrawals during the O.P.:
(a) Fiscal age last birthday
(b) Calendar fiscal age
CHAP1�ER 10
MULTI-STATE MODELS
JO.I INTRODUCTION
The models described in the first part of this text were characterized by a
primary variable of interest that was generally the time until failure (death)
of an entity (usually a living being) known to exist at some initial point.
Such models could be viewed as two-state models, in the sense that the
entity (person) is in one of two states, either alive or dead. Furthermore,
letting State A denote alive and State B denote dead, these models permitted
transfer from State A to State B only, as shown in Figure 10.1.
!FIGURE 10.1!
(Barring the mystical concept of resurrection, transfer from State B to State
A was not possible. In fact, there is no transfer out of State B at all.) Since
the decrement (transfer) from State A is death, then the hazard rate for such
transfer is called the force of mortality. In a more general setting, where
transfer can mean something other than death, the more general term force
of transfer will be used.
There are some early actuarial examples of multi-state models be­
yond the simple two-state case. For example, Jordan [41J presents a three­
state model in which persons in State A (active) can become disabled and
transfer to State B (disabled), according to an applicable force of disability.
Persons in State B can then die, thereby transferring to State C, according to
a force of mortality applicable to disabled lives. Of course persons in State
A can also die, governed by a force of mortality applicable to active lives,
thereby transferring to State C. Thus transfer in this model is possible from
State A to either State B or State C, and transfer is possible from State B to
State C. All disabilities in this model are assumed to be permanent, so trans-
248
1lfulti-State Models
fer from State B back to State A is not possible, and, of course, no transfer
from State C is possible. This model is illustrated in Figure 10.2.
!FIGURE 10.2!
In the terminology of Section 5.3, a person in State A of Jordan's
model is in a double-decrement environment, subject to decrements death (d)
and disability (i), and all of the functions defined in Section 5.3 would apply.
A person in State B of Jordan's model is in a single-decrement environment.
Several additional multi-state models are presented in this chapter,
along with, in some cases, procedures for estimating the model parameters
from sample data.
An interesting extension of the theory thus far presented in this text
arises in connection with multi-state models. Given the number of persons in
a particular state at some initial point of time (say, t = 0), then the number of
persons in that state at later time t in general is a consequence of the various
forces of transfer into and out of the state. The process leading to the number
in the state at time tis called a stochastic process. We will briefly examine
such a process in Section 10.3.4.
10.2 PROGRESSION OF A DISEASE
The first multi-state model we consider is one that models the progressive
stage of a disease, such as the AIDS model proposed by Panjer [59]. For the
convenience of readers who wish to pursue Panjer's original work, we will
adopt notation here that is consistent with [59].
10.2.1 Description of the Model
In Panjer's model, a total of six states are defined. A healthy person not in­
fected by the AIDS virus is said to be in State la, and a person infected by
the virus (i.e., testing HIV+) but having no symptoms of the disease is in
State lb. The next three states, denoted 2a, 2b, and 3, reflect progressive
249
Progression of a Disease
stages of the disease, and the final state is that of death. Without concerning
ourselves with medical meanings, we identify the three stages of the disease
as follows:
State 2a:
State 2b:
State 3:
Lymphadenopathy Syndrome (LAS)
AIDS-Related Compex (ARC)
Acquired Immune Deficiency Syndrome (AIDS)
This classification system is consistent with the Walter Reed Staging Meth­
od (see Redfield, et al. [62]).
It is assumed that persons progress through the successive stages of
the disease without returning to an earlier stage. For simplicity it is also as­
sumed that death prior to State 3, although certainly possible, is a risk small
enough to be ignored. This means that a person in any state can leave that
state only by progressing to the next state in the model (i.e., a single-decre­
ment environment), as illustrated in Figure 10.3.
State
la
Health
-+
State
State
State
State
State
-+
-+ 2a -+ 2b -+
3
4
lb
'Death'
HIV+
LAS
/ARC
AIDS
!FIGURE 10.3!
Since the only decrement in a given state is progressing to the next
state, then the model is eompletely defined by the hazard rates, which we
shall call forces of progression, operating in each state. A simplifying as­
sumption is that these forces of progression, denoted µj for j = Ia, lb, 2a,
2b, 3, are constant within each state, not depending on the age or sex of the
person nor the length of time the person has been in that particular state.
10.2.2 Properties of the Model
Since the force of progression is constant, then the random variable for
length of time spent in Statej, denoted Tj, is exponentially distributed, with
Srp) = Pr(Tj > t) = e- 1·µ1,
Fr/{) = Pr(Tj::; t) = I - e- 1·µ1,.
(IO.I)
(10.2)
( 10.3)
250
Multi-State Models
and
E[�] = Jj ,
(10.4)
I
2·
µj
(10.5)
Var(�)
Recall the "memoryless" property of the exponential distribution,
which says that � is the random variable for future time spent in State J,
whether measured from the time of entry into State j or any later time while
in State j. Furthermore, the random variable � is independent of the random
variables associated with the other states, so that progression to the next state
does not depend on how long the person was in a previous state.
The model assumes that persons leave State 3 (AIDS) only through
death, so µ3 is actually the force of mortality for persons with AIDS, and r3
is the future lifetime random variable for such persons. Then the random
variable
(10.6)
denotes the future lifetime of a person in State 2b (ARC), with expected
value
and
I
I
- µ1b + µ3
E[R]
(10.7)
(10.8)
due to the independence of T2h and T3 • Other random variables can similarly
be defined as sums of the basic random varialbles. For example, the random
variable
(10.9)
denotes the progression time from State lb (testing HIV+) to State 3 (AIDS),
with expected value
E[S]
(10.10)
Progression of a Disease
251
![EXAMPLE 10.t!I Suppose the expected future lifetimes of persons diag­
nosed with LAS, ARC, and AIDS are 6.14, 4.24, and 0.91, respectively.
Find the variance of future lifetime for persons diagnosed with LAS.
!SOLUTION! In the notation of Panjer's model, we have
I
I
I
µ2a + µ2b + µ3 = 6·14,
1
1 _ 4.24,
µ2b + µ3 and ...!.. = .91. From the given equations we easily find -1- = 3.33 and
µ2b
µ3
-1- = 1.90. Then the variance is given b y
µ2a
_I + _I_ + j_ = 15 53
2
2
2
· ·
µ2a
µ2b µ3
0
jlEXAMPLE 10.211 Find the probability that a person now in State i will be
in the next state t years from now.
!SOLUTION! Let the person move from State i to State i+I at time r, where
0 < r < t, and then remain in State i+I until time t. The (differential)
probability of this event is e-r·µ, • µ; · e-(t-r)µ;+i dr, so the total probability is
1
-r·µ; · µ; · e -(t-r)µ1+1 dr
Pu+1(t) = 1 e
0
= µ;. e-1·µ1+1 fo' erl.µ1+1-µ;) dr
= µ;. e-1-µ1+1 [et(µ;+i-µ,) µ;+) - µ;
I]
(10.l la)
(10.l lb)
0
Multi-State Models
252
In the special case where State i + 1 is death, from which there is no
further transfer, then µ;+ 1 = 0 and the result of Example 10.2 reduces to
1 - e-t·µ;, the probability of death within t years while in State i.
The general result of Example 10.2 can be extended to find the prob­
ability of progressing from any state to any later state and remaining in that
later state to time t. For example, consider the probability of moving from
State i through State j and on to State k (but no further) by time t. If the
transfer from State i to State j occurs at time r, 0 < r < t, and the transfer
from State j to State k occurs at times, r < s < t, then the desired probabil­
ity is given by
=
=
-- - - ]
=
µ; . µJ e-1·µ;
[
µj
µk - µj
=
e-t·µ;(µk-µj)
µi. µ i
µk - /J,j ({Lj-µ;)(µk-µ;)
[
e-1·µ1
/J,;
e-t·µ;
e-t·µk
/J,k - /J,;
e-1·µ1
+ e-1·µ1
µj - µ;
µk - /J,;
]
=
(10.12)
253
Progression of a Disease
It is easy to see that Equations ( I0.11b) and (I0.12) are of the same form.
Then in the general case, the probability of moving from State i, through
State j,.. . , to State n, and remaining in State n to time t, is given by
(10.13)
IIEXAMPLE 10.41) Using the forces of progression developed in Example
10.1, find the probability that a person with LAS will die within 5 years.
!SOLUTION! In the notation of Panjer's model, we seek the value of
p2a,2b,3,4 (5), the probability of progressing from State 2a, through State 2b, on
through State 3, and into State 4 by time t = 5. The answer is given by
Equation (10.13), withµ;=µa 2 = .526,µj =µ2b = .300,µk = µ3 = 1.099,
and µn = µ4 = 0, since State 4 is death. Then we have
_-5(.526)
P2a,2b,3,4( 5) = (-526)(.300)(I.099) [ (0 .526)(1. 099-.526)(.300-.526)
e-5(.300)
0-.300)(1.099-.300) .526-.300)
+ �����������(
(
e-5(1.099)
+ (0-1.099)(.300- I .099)(.526- I .099)
e-5(0J
+ ( l .099-0)(.300-0)(.526-0)
.004
.223
= · 173 [ .072
_I_] _ .470 .
.068 + -.054 + -.503 + .173 -
l
0
254
Multi-State Models
Returning to Equation (10.6), let us now find the exact distribution
of the random variable R = T2b + T3, the future lifetime (or time until death)
of a person diagnosed with ARC. The CDF of R, denoted FR(t) = Fr1b+r,(t),
is the probability that death will occur by time t for a person now in State 2b.
This probability is given by Equation (I 0.12), with i replaced by 2b, j
replaced by 3, and k replaced by 4 (which is death), so that µk = 0. Then we
have
b
3
µ3 . e-t·µ.z
µ2b . e-t·µ.
= ----+
+l
)
)
- (µ2b-µ3
·µ zb
µ3 - µ2b + µ2b · e-t·µ.3 - µ3 · e-t .
- (µ3-µ2b
=
µ3-µ2b
µ2b(l-e-t·µ.3)
= µ3(1-e-t·µ.zb) +----µ3 - µ2b
µ2b - µ3
µ3
µ2b
= --· Frit).
· Fr2/t) +
µ3 - µ2b
µ2b - µ3
(10.14)
The survival function of R = T2b + T3 is
= 1 - Fr2b +r/t)
= µ3 . Srlb(t) +
µ3 - µ2b
µ2b .
Sr3(t),
µ2b - µ3
(10 .15)
and the PDF is
fr2b+r/t) =
=
(10.16)
The extension to Equation (10.9), or to any other case of the sum of three or
more basic random variables, is easily seen.
10.2.3 Available Data
Because of the assumption of constant forces of progression, Panjer' s model
is quantified by estimating the values of µj, for j = 1a, 1b, 2a, 2b, 3. As
demonstrated several times in Part II of this text (see Equations (7.15) and
(7.23)), if we have sample data showing the exact time under observation at
Progression of a Disease
255
which sample members progressed to the next state, then we can easily
calculate exact exposure and find the maximum likelihood estimate of each
constant force of progression as the ratio of number of progressions to exact
exposure.
In practice, however, it is not always the case that sample data exists
in such an ideal form, so that the researcher is faced with the issue of how
best to adapt the estimation procedure to the form of available data. This
was true at the time of Panjer's study. Panjer used data that was available
from a study done at the University of Frankfurt in Germany.
The Frankfurt data recorded the following three pieces of informa­
tion about each person in the study:
(1) The length of time the person was under observation.
(2) The stage the person was in at the beginning of observation (the
initial stage).
(3) Whether the person remained in the initial stage throughout ob­
servation, or left that stage while under observation.
Note that item (3) only reports whether or not the person left the initial stage
while under observation. For those who did leave, the data does not show
whether these persons progressed only to the next stage or to several later
stages.
The total number of persons in the Frankfurt study was 543. Of this
total, however, 236 were under observation for less than three months, and
were not considered further. The remaining 307 constituted the sample for
Panjer's estimation work. Of this 307, 48 were at risk but not infected (State
la) and the other 259 were infected at the time of initial observation. Of the
259 infected persons, 52 were lllV+ but otherwise asymptomatic (State lb),
120 were diagnosed with LAS (State 2a), 64 were diagnosed with ARC
(State 2b), and 23 were diagnosed with AIDS (State 3).
Although the original data may have shown the precise length of
time under observation for each person, the published data does not. Rather
it groups the 307 sample members into four time-under-observation groups:
those under observation for 3 to 6 months (54 persons), those under obser­
vation for 6 to 12 months (121 persons), those under observation for 12 to 24
months (97 persons), and those under observation for 24 to 36 months (35
persons).1
!The interested reader can find more detail on the Frankfurt data in Cowell and Hos­
kins [20 ].
256
Multi-State Models
10.2.4 Estimating Model Parametersfrom Sample Data
The parameters µj, forj = Ia, I b, 2a, 2b, 3, are estimated separately from the
data described in Section 10.2.3 by the maximum likelihood approach. For
the fh initial stage, recall that the Frankfurt data shows, for each of four
time-under-observation groups, the total number in that group (n;) and the
number (subset) of them who progressed out (d;) while under observation,
where n; and d; refer to the i1h time-under-observation group. Then n;-d;
denotes the number who remained in the initial stage throughout observa­
tion.
Let Pi denote the probability of remaining in the initial stage and
q; = 1-p; denote the probability of progressing out of the initial stage. It is
clear that we have a binomial model here, so the likelihood for estimating µ1
IS
(10.17)
for the i1h time-under-observation group and
4
L(µj) = IJ(l-p;)d;. (p;r-d,,
(10.18)
i=I
since there are four time-under-observation groups. The log-likelihood is
f(µj) =
L d; · log(l-p;) + (n;-d;) · logp;
4
i=l
(10.19)
and the likelihood equation.is
= 0,
(10.20)
which can be solved for f),1 by iteration.
Of course we first need to express p; in terms of µj. Since µ1 is
assumed constant, then p; = e-r,µj, where r; denotes the length of time
remaining under observation in the j'h stage (without progressing out) for
persons in the i'h time-under-observation group. For convenience we assume
that each person is observed for half of the maximum possible observation
Progression ofa Disease
257
time in each group. Thus for the first group (3 to 6 months), we take ri to be
4.5 months, or f year. For the second group (6 to 12 months), we take ri to
year. For the third group ( 12 to 24 months), r; is 18
be 9 months, or
months, or years, and for the fourth group (24 to 36 months), r; is 30
months, or � years. This assumption now allows us to express the likelihood
for µj in terms of µj, as illustrated in the following example.
f
f
!!EXAMPLE 10.51! Using the Frankfurt data and Panjer's approach, show
how to estimate µ1b, the force of progressing out of stage for persons diag­
nosed HIV+.
!SOLUTION! The date gives the numbers of persons in this stage when first
observed to be n1 = 9, n2 = 18, n3 = 20, n4 = 5, for the four time-under­
observation groups. The numbers progressing out of this stage while under
observation are d 1 = 1, d2 = 10, d 3 = 15, d 4 = 4. Then the likelihood
function is
Substituting for eachp; in terms of µ1b we have
) b
) 1b] 1 . (e-(3/8µ
/ µ
1 ]8. (l -e-(3/4)µ1b]10. [e-( 3/4)µ1bs]
L(µib) = (1-e-(J8
. (l-e-( 3/2)µ1bJ15. [e-( 312)µ1bJ5. [l-e-(S/2)µ1bJ4. [e-(S/2)µ1b]1.
(µ,b) = 0 is then solved by iteration, or the like­
The likelihood equation 88£µ1
b
lihood L(µ1b) can itse lf be maximized by computer routine. In this ;ease the
D
solution turns out to be µ 1b = .86.
As an alternative to using exactly half of the maximum time under
observation for each person, Panjer investigated the idea of using a random
censoring mechanism for assigning times under observation to each person
in the sample, as described in Section 7.3.5. Thus for the first time-under­
observation group of 3 to 6 months, for example, instead of using 4.5 months
for every person, the respective times under observation were generated by a
random process from a uniform distribution on (3,6). The random censoring
alternative produced results nearly identical to those produced by the regular
approach.
258
Multi-State Models
10.2.5 Analysis of the Estimators
µ
The asymptotic variance of the MLE of µ1 , 1, is given by
(10.21)
which is then numerically estimated as
(10.22a)
by substituting the estimate µ1 for µ1 in Equation (I 0.21 ). Alternatively, in
this model the same estimated variance is obtained as
2
Var(µ'J) = {- {) lnL
aµJ
}-1
(10.22b)
without first obtaining the expected value. The eq uivalence of Equations
( l 0.22a) and (10.22b) is demonstrated in Appendix D.
[f the estimator can be assumed to be asymptotically normal (see
Appendix A), then confidence intervals for the estimation results can be
easily established. The interested reader is referred to Panjer's original work
[59] for a presentation of the numerical results of this, as well as other,
summary measures of his research.
10.3 MODELING CONTINUING CARE
RETIREMENT COMMUNITY POPULATIONS
A convenient feature of the AIDS model of Section 10.2 was that a person
could move only "from left to right," reflecting the progress of the disease. In
this section we consider a more complex stochastic model, one that analyzes
the survival pattern of the residents of a continuing care retirement
community (CCRC). The model presented in this section was proposed by
Jones (40]. Interested readers should consult this reference for a fuller
discussion of this model.
Modeling Continuing Care Retirement Community �opulations
259
J0.3.1 Description of the Model
The CCRC is made up of m independent living units (ILU) and a skilled
nursing facility (SNF), to which residents can transfer on either a temporary
or permanent basis. Transfer to the SNF is considered temporary if the
resident is expected to recover and return to his or her ILU, which will
remain vacant during the temporary stay at the SNF. If a return to inde­
pendent living is not expected, then the transfer to the SNF is considered
permanent. Residents may leave the community, either by death or voluntary
withdrawal, at any time while residing in their ILU's or in the SNF (either
temporarily or permanently).
Thus we see that there are four states to the model:
State (a):
State (b):
State (c):
State (d):
Residence in the ILU.
Temporary residence in the SNF.
Permanent residence in the SNF.
Departed from the CCRC, either by death or voluntary
withdrawal.
Persons in State (a) can transfer to any of States (b), (c), or (d). Persons in
State (b) can transfer back to State (a), or to States (c) or (d). Persons in
State (c) can transfer only to State (d), not back to States (a) or (b). Persons
in State (d) are departed from the CCRC, and cannot transfer to any of the
other states. The model is illustrated in Figure 10.4.
(ILU)
State (b)
(Temporary
SNF)
State (d)
(Departed from
the CCRC)
State (c)
(Permanent
SNF)
State (a)
1
1
!FIGURE 10.4!
260
Multi-State Models
10.3.2 Assumptions and Properties of the Model
It is assumed that there is an adequate waiting list for residence in the
CCRC, such that any vacancy occurring in an ILU as a result of transfer to
either State (c) or State (d) is immediately filled by a new resident. As stated
above, transfer to State (b) keeps the ILU vacant pending transfer back to
State (a) upon recovery. However, if the person considered to be tempor­
arily in the SNF (i.e., in State (b)) is reclassified as a permanent resident of
the SNF or dies or otherwise withdraws (i.e., transfers to State (c) or State
(d)), then the vacancy in that person's ILU is immediately filled.
This "high demand" assumption has two important consequences
The first is that the total number of ILU residents plus temporary SNF
residents ism at all times. The second is that them ILU's can be assumed to
operate independently of each olher. This, in turn, allows us to analyze the
model by first analyzing a single unit.
10.3.3 Transition Probabilities
As with the Panjer AIDS model of Section 10.2, we assume here that the
force of transition from State (i) to State (j), denoted µif, is constant. Thus
µab denotes the force of transition for a resident in an ILU to make a tem­
porary transfer to the SNF. Similarly, µba denotes the "force of recovery" for
a resident in the SNF to return to an active status in his or her ILU.
As explained in Section I 0.3.1, and illustrated in Figure 10.4, the
non-zero forces of transition for a person in State (a) are µab, µac, and /J,ad·
The non-zero forces for a person in State (b) are µha, µbe, and µbd· For a
person in State (c), the only non-zero force of transition is µcd. It should be
clear that no forces of transition apply to persons in State (d), since all such
persons have departed from the CCRC.
From these observations it is clear that a person in State (a) is facing
a triple-decrement environment, so that various probabilities can be deter­
mined from the three forces of transition using familiar actuarial concepts
and notation. In particular, the total force of transition is given by
(10.23)
Since this total force is constant, then the random v�riable for length of time
spent in State (a), denoted Ta , has an exponential distribution. Then the sur·
vival function for a person in State (a) at time t = 0 is given by
Modeling Continuing Care Retirement Community Populations
261
The expected length of time to be spent in State (a) is given by
( I 0.25)
and the variance is given by
(10.26)
Similar remarks apply to Tb, the random variable for length of time
spent in State (b), a temporary stay in the SNF, with total force of transition
given by
(10.27)
Persons in State (c), however, are subject only to the force of transition to
State (d), a single-decrement environment. The random variable Tc has an
exponential distribution with survival function
(10.28)
Because of the "memoryless" property of the exponential distribu­
tion, survival probabilities in any state are independent of the length of time
a person has been in that particular state. Clearly the assumption of constant
forces of transition, with its consequent exponential probabilities, is ques­
tionable. It has been made here mainly to simplify the analysis.
IIEXAMPLE 10.6!j For a person currently residing in an ILU, give expres­
sions for each of the following:
(a) The probability of not leaving the ILU at all during the next year.
(b) The probability of leaving the ILU by reason of permanent trans­
fer to the SNF during the next year.
(c) The probability of leaving the ILU be reason of temporary
transfer to the SNF during the next year, and continuing to be a
temporary resident of the SNF at the end of such year.
262
Multi-State Models
lfili!J}TION!
(a) This probability is given directly by Sa(l) = e-µ�.
(b) The force of permanent transfer to the SNF (State (c)) is denoted µac,
The desired probability is given by
(c) This is an extension of part (b), where the event was simply transfer to
State (c). Here the event is transfer to State (b) and survival in State (b)
until time t = 1. The probability is given by
The similarity of this result to Equation (10.11b), developed in connec­
tion with Panjer's AIDS model, should be noted.
0
10.3.4 Stochastic Processes
Part (c) of Example 10.6 is itself a small part of a larger question that we
will address in this section. The need for future actuarial analysis of CCRC's
is likely to be quite significant. Jones [40] points out that in order to carry
out this actuarial analysis, we will need an appropriate model that describes
a CCRC population over time. He also observes that "Such a model is
necessarily rather complex."2
The general question is as follows: If a CCRC has m fully-occupied
independent living units (ILU's) at some initial time point, denoted t = 0,
subject to the "high demand" assumption described in Section 10.3.2, then
2see (40], p. 433.
Modeling Continuing Care Retirement Community Populations
263
what is the joint probability distribution of the number of temporary SNF
residents and the number of permanent SNF residents at time t in general?
Let J(t) denote the number of permanent SNF residents at time
t > 0. For a fixed value of t, J(t) is a random variable with a probability
distribution. The collection of the random variables J(t) produced by various
values oft is called a stochastic process. If t takes on only a countable set of
points, then the process is called a discrete-time stochastic process, and if t
takes on all values in a continuous set of points, then the process is called a
continuous-time stochastic process. Notationally, the process is distin­
guished from its member random variables by writing { J(t); t E T}. When the
process is a continuous-time process, which is the case here, we write
{J(t);t > 0}.3
Similarly, let K(t) denote the number of temporary SNF residents at
time t > 0, with associated continuous-time stochastic process denoted
{K(t); t > O}. Then the actuarial analysis of CCRC's proposed by Jones
addresses the bivariate stochastic process {J(t), K(t); t > O}.
Jones assumes that the stochastic behavior of any particular ILU is
independent of that of any other ILU. This assumption permits us to consider
first the process associated with one arbitrary ILU, say Unit l, and to later
obtain information about {J(t), K(t); t > O} from the information known
about the Unit 1 process, denoted {J1(t), K1(t); t > O}.
What do we know about J1(t) and K 1 (t)? First we note that K1 (t) is
the number of temporary residents in the SNF arising only from ILU l, so it
can only take on the values O and 1. (It is assumed that K 1 (0) = 0.) There are
many mutually exclusive ways by which Ki (t) could be zero at the end of the
time interval (0, t). For example, we might have had no one transferring from
the ILU (State (a)) to the temporary SNF (State (b)) in (0, t) at all. Or a
person could have so transferred, but then transferred back to State (a)
before time t. Or a person could have so transferred, but then died, thereby
transferring to State (d), before time t. Or a person could have transferred
into and back out of State (b) several times prior to time t, with a net result
of no one in State (b) at time t. Thus it is easy to see that the overall
probability that K 1 (t) = 0 is not easily obtained.
Similarly, there are many ways to end up with the result K1 (t) = 1.
(Part (c) of Example 10.6 presented one such way.) It can be shown that this
probability is given by
3for more on stochastic processes, see Chapter 3 of Panjer and Willmot [60].
264
Multi-State Models
(10.29)
a result derived by Ross [65]. Then the probability that K1 (t) = 0 is given
by
Pr[K1(t) = Oj K1(0) = O] = l - Pr[K1(t) = I I K1(0) = 0]
= µ'i, + µab . e-(µ.b+µi)t
µab+ µ'i,
(10.30)
and (l 0.29) and (I 0.30) together give the (marginal) distribution of Ki(t).
The random variable J1 (t) can take on any integer value t = 0, 1, 2,
... , and its marginal distribution is considerably more complicated than that
of K1(t)
Rather than pursue the actual distributions of J1 (t) and K1(t), Jones
shows how to obtain expressions for E[J1 (t)], E[K 1 (t)], Var[Ji(t)], and
Var[K 1 (t)], as well as Cov[J 1 (t), J 1 (t)].4 Then by the assumption of mutual
independence and identical distribution of all ILU's, we can find
and
E[J(t)] = m · E[Ji(t)],
(l 0.3 l)
E[K(t)l = m · E[K1(t)],
(10.32)
Var[J(t)] = m · Var[J1 (t)],
(10.33)
Var[K(t)] = m · Var[K 1 (t)].
(10.34)
Finally, if the number of ILU's in the CCRC is sufficiently large, it will be
reasonable to approximate the distributions of J(t) and K(t) by nonnal distri­
butions, which are completely defined by the first two moments given by
Equations (10.31) through (10.34).
The total number of residents in the SNF, whether temporary or
permanent, is given by S(t) = J(t)+K(t). The distribution of S(t) can be ap­
proximated by a normal distribution with parameters
and
E[S(t)] = E[J(t)] + E[K(t)],
(10.35)
4Thcse derivations are beyond the scope of this text the interested reader is referred
to Jones [40].
Exercises
265
Var[S(t)] = Var[J(t)] + Var[K(t)] + 2 · Cov[J(t), K(t)],
(10.36)
Cov[J(t), K(t)] = m · Cov[J1 (t), K 1 (t)].
(l 0.37)
where
10.4 EXERCISES
10.1 Introduction; 10.2 Progression of a Disease
I0-1. Calculate the life expectancy for a person diagnosed with LA S in
Panjer's model, given the following information.
(i) µ2a = .50
(ii) The variance of future lifetime for a person in State 2a is 6.60.
(iii)The expected future lifetime for a person in State 3 is .80.
10-2. In Panjer's model, find the PDF of the random variable R = T2b + T3 ,
given the following information.
(i) µ2b = t
(ii) The life expectancy for a person in State 2b is
�6.
10-3. Find, in terms of µ1b and µ2a, the probability that a person known to
be in State lb at time t = 7 will be in State 2a at time t = 15.
I 0-4. Suppose 5 persons are all in State j of a Panjer-type model when they
come under observation. Three of them are observed for exactly two
years, during which time two of them progress to State j+1 and the
other remains in State j. The other two persons are observed for
exactly one year, during which time one progresses to Statej+ I and
the other remains in State j. Using Panjer's methodology, find the
equation whose solution yields the MLE of µ1.
10-5. Using Panjer's methodology, find the probability of progressing out
of State l a if under observation for 3-6 months, given the following
information.
(i) µ1 0 = 3k
(ii) µ2b = k
(ii) The probability of progressing out of State 2b if under observa­
tion for 12-24 months is .36.
Mulfi-State Models
266
l0-6. Using Panjer's methodology, and the following data, find the MLE of
µ3.
Time under
Observation
6-12 months
12-24 months
24-36 months
Number
of AIDS
Patients
12
7
1
Number
of Deaths
Observed
8
5
0
l 0-7
Suppose 100 persons come under observation while in State j of a
Panjer-type model, and all remain under observation for one year.
Let µj denote the MLE of the constant force of progression µj. Find
an expression for the asymptotic variance of µj .
10.3
Modeling Continuing Care Retirement Community Populations
10-8. Using the notation and assumptions of Jones' model, find expressions
for each ofthe following.
(a) The probability that an ILU resident will never occupy the SNF
on a temporary basis.
(b) The expected length ofa single stay in the SNF.
(c) The expected number of residents to occupy a given ILU during
the next ten years, given that the current resident is now in State
(a).
CHAPTER II
APPLICATIONS IN ECONOMICS AND FINANCE
II.I INTRODUCTION
In this chapter we describe three examples of applications of survival mod­
els, and the accompanying exercise of estimating those models from sample
data, that arise in the disciplines of economics and finance. The first applica­
tion deals with modeling and estimating the duration of some economic
event, such as the length of a spell of unemployment or the length of a labor
strike. Also included are brief examinations of the survival patterns of con­
ventional mortgages and corporate bonds.
11.2 DURATION OF ECONOMIC EVENTS
There are many examples of important durations that arise in economic or
financial analysis. Specific areas that have been studied include duration of
marriages, spacing of births, time between job moves, duration of wars, time
in office, duration of legal cases (from initiation to resolution}, time in
military rank, and length of stay in graduate school. Two other very
important economic durations, length of a spell of unemployment and length
of a labor strike, are examined in this section.
There is an extensive literature on the topic of estimation of eco­
nomic durations. Much of the material in this section is based on Kiefer
(47], which itself has an extensive bibliography for further research. The
specific issue of unemployment duration, along with a comprehensive survey
of the literature on that topic, is addressed by Kiefer and Devine[48].
11.2.1 Duration of Unemployment
The importance of a reliable predictive model for length of time spent in un­
employment should be self-evident. Advance estimates of premiums for un­
employment insurance coverages depend heavily on expected durations of
unemployment. Additionally, the distribution of the duration of unemploy­
ment plays a critical role in economic theories of job search, so that any
careful evaluation of such theories requires accurate information on the dura­
tion of unemployment.
268
Applications in Economics and Finance
Some information on the distribution of unemployment duration is
available from the reguiarly-scheduled monthly Current Population Survey,
but this data has serious shortcomings. In this survey, persons unemployed
at the time of the survey are asked how long they have been unemployed.
This means that data on persons who are employed at the time of the survey,
but who may have been recently unemployed, is not obtained. The problem
is illustrated on the following figure.
Case l
Case2
--+-------·?
Survey
Date A
!FIGURE 11.1
!
Survey
DateB
If monthly surveys of unemployed persons are done on Date A and
Date B, then any spells of unemployment such as that denoted Case 1 in
Figure 11. l will not be reported. Since any unreported cases of this nature
are necessarily less than one month in duration, then this approach will seri­
ously underreport spells of short duration, a phenomenon known as lengtli­
hiased sampling.
A second consequence of sampling only the unemployed is that they
can report only the lengths of their spells of unemployment to t.iate, resulting
in data that has been right-censored. This is illustrated by Case 2 in Figure
11.1 in which only a fraction of the actual length of unemployment would be
reported on Survey Date A.
To avoid both the right-censoring and the length-biased sampling in­
herent in the Current Population Survey, some surveys do sample employed
as well as unemployed persons as of some date (such as Survey Date B), and
ask if the respondent has been unemployed at any time since a specified
prior date (such as Survey Date A). Under this approach, spells such as Case
1 will be accurately reported, as will the otherwise right-censored portion of
Case 2. However this approach might record only the duration of Case 2
since Survey Date A, thereby underreporting the portion of it prior to Survey
Date A. In this case the spell has been left-censored.
Duration ofEconomic Events
269
11.2.2 Model Selection
As with any other example of analyzing a future lifetime random variable,
the analysis of economic durations, such as length of unemployment, re­
quires a choice of model for the distribution of the primary variable. Further­
more, as we have seen throughout this text, the distribution of a survival time
variable can be expressed in terms of the survival function, S(t), the distri­
bution function, F(t), the density function,f(t}, or the hazard function, .X(t).
A review of the economic literature on this subject reveals a preference for
using the hazard function approach, with its conditional probability nature,
over the (unconditional) density function approach.
!!EXAMPLE 11.tl! Suppose an unemployed worker spends a fixed amount
of time each day looking for employment, with a constant probability, .X, of
finding employment on any given day. What is the probability that the spell
of unemployment will end on its tenth day?
!SOLUTION! The answer is given by the geometric distribution, a special
case of the negative binomial, as
/(10) = .X(l-).)9 •
D
The extensive use of hazard functions in economic duration analysis
influences the selection of the model for the distribution. In particular, if our
theoretical notions about an economic phenomenon suggest that the hazard
rate might be constant over time, then we would avoid using the normal or
lognormal distributions, which have complex hazard functions that do not
admit the constant hazard as a special case, in favor of distributions such as
exponential and Weibull, which easily accommodate the constant hazard case.
Given that an economic event has existed for length of time t, the
hazard function at time t, .X(t), measures the likelihood that the event will
end soon after time t. If the hazard function is increasing, indicated by
).(t) > 0, then positive duration dependence is said to exist. This means
that the probability that the event will end soon increases as the length of the
event increases. In the unemployment duration example, positive duration
dependence means that the longer the unemployment has lasted, the more
likely it is that it will end soon, due, perhaps, to increased job search in­
tensity or a lowering of the minimum acceptable wage. Conversely, if
.X(t) < 0, so that the hazard function is decreasing, then we have negative
duration dependence. In this case the longer the unemployment has per­
sisted, the less likely it is to end soon, due, perhaps, to near exhaustion of
i
i
Applications in Economics and Finance
270
employment opportunities. Finally, if the hazard function is constant, then
we have no duration dependence at all, which means that the likelihood of
having the unemployment end soon is independent of how long it has per­
sisted. This leads to the familiar exponential distribution, with its "memory­
less" property.
Although the exponential distribution is easy to work with, the fact
that it depends on only one parameter limits its flexibility. This is a draw­
back in economic duration applications, especially if the data contain both
very long and very short durations. An alternative to the exponential for use
with economic duration data is the two-parameter Weibull distribution,
given by Equation (2.35) as
= k·x n ,x 2: 0, k > 0,n > -1.
(2.35)
Making the parameter substitutions k = 0./3 and n = /3-1, along with replac­
>.(x)
ing x with t (which is more suitable for durational models), we have
>.(t) = a/3 · tf3-1,t 2: 0,o. > 0,/3 > 0.
(11.1)
The survival function of the Weibull distribution was given by Equation
(2.36). Making the parametric substitutions above we have
S(t) = e-0·1 .
(11.2}
13
Similarly, the density function is
f(t) = - $iS(t) = S(t) · >.(t) = a/3 · tf)-1 • e-0·1 ,
13
(11.3)
which was developed earlier in Example 8.5.
!!EXAMPLE 11.21! Show that if (3 = I, then the Weibull distribution re­
duces to the exponential distribution.
!SOLUTION! If /3 = l, then we have
S(t) = e -at '
and
f(t) = a·e-01,
>.(t) = Ci,
(ll.4a)
(ll.4b)
(ll.4c)
a constant. Clearly we have the exponential distribution given by Equations
0
(2.26), (2.27), and (2.28), respectively,with a replaced by .X.
Duratio11 ofEconomic Events
271
If the time axis is rescaled using the transformation of duration
given by
s = 1/3 '
(11.5)
where T has a Weibull distribution, then S = T/3 has an exponential distribu­
tion. To show this result, note first that f3 > 0 so tf3 is a strictly increasing
function and the inverse function t = s 11f3 exists, as required by Section 2.5.
Then by Equation (2.64) we have
(11.6)
the SDF of the exponential distribution of S.
To analyze the duration dependence of the Weibull distribution
given by Equation (11.1), we first find
3t>.(t) = 3t(o:f3·tf3-I) = o:(3((3-I)·t(J-2.
(11.7)
If /3 > 1, then �>.(t) > 0, and we have pos-itive duration dependence; if
{3 < 1, then f�(t) < 0, and we have negative duration dependence; and if
(3 = l, then j>.(t) = 0, and we have no duration dependence, as already
known since f3 = 1 reduces the Weibull distribution to the exponential.
!!EXAMPLE 11.311 Show that the duration dependence of the Weibull distri­
bution does not depend on the value of the parameter a:.
ISOLUTION! Since o: > 0, the right side of Equation (11. 7) will be greater
than, equal to, or less than, zero according to f3 � I, regardless of the value
0
of a:.
The two-parameter log-logistic distribution, defined by
-y6 · 1 ° - 1
>.(t) =
6, t 2: 0, -y > 0, 6 > 0,
1 +,·t
(ll.8)
has a nonmonotonic hazard function if 6 > 1, first increasing and then
decreasing with duration. The interpretation of this in the unemployment
duration example is that the likelihood of finding employment in the near
future grows as the length of time unemployed increases up to a point, but as
unemployment persists beyond that point it then becomes less likely that
employment will soon be found. This, and other properties of the log-logistic
distribution, will be pursued in the exercises.
Applications in Economics and Finance
272
1 J.2.3 Use of Economic Tlteory
To assist us in choosing a distribution for our length of unemployment ran­
dom variable, we tum to the economic theory of job-search models. The
objective is to have the economic theory suggest the nature of the hazard
function, so that we can then choose a distribution whose hazard function
possesses such a nature.
Let us consider that we have a simple two-state model, as described
in the introduction to Chapter I 0, where State U is unemployment and State
E is employment. We will focus on the force of transition from State U to
State E, which, of course, is the hazard function of the random variable for
length of unemployment.
In this model, we assume that an unemployed person receives em­
ployment offers at a constant rate 'f/, so that the probability of receiving an
offer in any short interval of length .:1 is rJ.i:1. The wage, w, associated with
each offer is subject to the distributionp(w).
The decision of whether or not the unemployed person accepts the
offer will depend on the utility associated with unemployment versus the
utility associated with employment. In this model we assume that the utility
associated with unemployment, denoted v11 , is constant over the spell of
unemployment and does not depend on the wage offer w. On the other hand,
we assume that the utility associated with employment, denoted ve (w), is an
increasing function ofw (i.e., higher wages are preferred to lower ones).l
Given these assumptions, it is possible to determine a person's
reservation wage, denoted w*, the wage at which he or she is indifferent
between accepting or declining the offer. Then since ve (w) is an increasing
function of w, the person will prefer employment at all wages greater than
w*. Then the probability that an offer is accepted is the same as the
probability that the offer is at wage w* or higher, which is given by
00
7f
= J. p(w)dw.
w•
(11.9)
Finally, the force of transition from unemployment to employment is given
by
(11.10)
>, = rJ7r,
the product of the offer rate and the probability that the offer is accepted.
Since this force of transition (hazard rate) is constant over the duration of
unemployment, we see that this particular economic theory suggests that an
exponential distribution for the length of unemployment is appropriate.
I For more on utility theory, see Chapter l of Bowers, et al. [12].
Duration ofEcohomic Events
273
A different conclusion might be reached if we reverse the states and
consider the force of transition from employment to unemployment as a
result of layoff. In this case it is reasonable to assume that layoffs occur in
inverse order of seniority, so that the hazard function for the random variable
length of employment would decline with employment duration, and an
exponential distribution would not be appropriate. Instead we might choose
a Weibull distribution with f3 < 1.
11.2.4 Non-Parametric Estimation
The data given in Table II.I, taken from Kennan (45], show lengths of
strikes in U.S. manufacturing industries over the period from 1968 to 1976.
To achieve a degree of homogeneity, only official strikes involving 1000 or
more workers, in which the major issue was general wage changes as class­
ified by the Bureau of Labor Statistics, are included. Furthermore, only
strikes beginning in June of each year in the study period are considered to
avoid seasonal heterogeneity. There are 62 total strikes in the data set.
There are 12 strikes in the data set whose lengths were right­
censored at 80 days. There are also several ties in the data, with 4 strikes
lasting 2 days, another 4 lasting 3 · days, and so on.
For each ordered duration j, let tj denote the length of that duration
and let <4 denote · the number of strikes lasting exactly lj days, so that · <4
represents the number of strike terminations at time ;. Let rj denote the risk
set of strikes still in effect just before time IJ· Then the conditional probabili­
ty of ending a strike at duration tj , given that it has reached duration tj , is
estimated by
d·
A
(I I.I la)
r··
qj - ':}_
J
The literature dealing with the duration of economic events tends to refer to
this conditional (discrete) probability as a hazard function, although in this
text we tend to restrict the term hazard function to the continuous case.
Then we can also write
A
d·
(11.1 lb)
,\(tj) = 'if.
J
The natural estimator for the survival function is the Kaplan-Meier
(or product-limit) estimator, developed in Section 7.6, which is
s<;) = fl (r;-;_d;).
i=I
I
(11.12)
Greenwood's formula (see Section 7.6.3) can then be used to estimate the
A
variance of the estimator S(lj).
274
Applications in Economics and Finance
Ordered
Duration
Number j
1
2
3
4
5
6
7
8
9
Duration
in Days
TABLE 11.1
lj
dj
Tj
1
2
3
4
1
4
4
1
1
62
61
57
53
52
51
5
7
8
9
I
1
2
l
l
2
1
50
Hazard
Survivor
.016
.066
.070
.019
.019
.020
.020
.041
.021
.022
.044
.023
.024
.024
.025
.026
.053
.028
.029
.029
.030
.063
.033
.034
.036
.037
.038
.040
.042
.043
.045
.095
.053
.984
.919
.855
.839
.823
.806
.790
.758
.742
.726
.694
.677
.661
.645
.629
.613
.581
.565
.548
.532
.516
.484
.468
.452
.435
.419
.403
.387
.371
.355
.339
.306
.290
· .258
.226
.210
.194*
/\
.X. (tj)
49
47
IO
IO
11
46
11
12
45
12
43
13
13
l
14
42
14
41
15
1
15
17
40
l
16
l
39
19
17
2
21
38
18
l
22
36
19
35
1
23
20
25
1
34
21
l
33
26
32
22
2
27
30
23
1
28
29
24
l
29
28
25
32
I
27
26
33
l
27
26
1
35
28
l
25
37
29
I
24
38
30
23
41
1
22
42
31
1
2
21
32
43
1
19
44
33
. II I
2
18
49
34
35
16
52
2
.125
14
.071
36
1
61
.077
72
13
37
1
•Twelve observations are censored at duration 80 days.
Source: Kennan (45), Table 1
s (tj)
/\
Duration ofEconomic Events
275
IEXAMPLE 11.4!! Use Greenwood's formula to estimate the variance of
the Kaplan-Meier estimator ofS(16) for the strike data shown in Table 11.1.
!SOLUTION! Using Equation (7.73) we have
Var[S(l6) I {r1}] � [S(l6)] 2 ·
.,.._
I\
d'i
L
J=I r/r1-<4)
15
(.645)2 [.0002644+.0011504 + · · · + .0006098]
= .00369.
0
The estimated hazard function and the estimated survival function are
displayed in Table II.I, and the estimated survival function is graphed in
Figure 11.2. Note the effect of the censored data on the graph ofS(l_j). The
estimated hazard function is graphed in Figure 11.3, and shows an implausi­
ble amount of variation over time. (A natural response by an actuary or an
economist would be to smooth out the graph by graduation.) A more useful
graph is that of the cumulative hazard function (CHF), defined in Chapter 2
as
A(t) = - lnS(t),
(1l.13a)
or, in the discrete analogue,
(l I.13b)
to be estimated as
(l 1.13c)
This is illustrated in Figure 11.4.
To interpret Figure 11.4, recall that for the exponential distribution
>.(t) = >., a constant, so the cumulative hazard function is
A(t) =
fo' >.(y) dy = >.t,
(11.14)
a linear function of duration. Figure 11.4 is close enough to linear to suggest
that the exponential model may be an appropriate parametric representation
of this data. Conversely, a convex cumulative hazard implies an increasing
hazard function (positive duration dependence), whereas a concave cumula­
tive hazard implies a decreasing hazard function (negative duration depen­
dence).
276
Applications in Economics and Finance
Survivors
0.1
0.0-1--��-,������������,������,���
O
IO
20
30
40
50
Duration in Days
60
70
80
!FIGURE 11.2!
Hazard
.12.11.10.09.08.07.06.05.04.0J­
.02.01.00-1--�����-,
���,������,���������,
,
1'o
80
O
IO
z'o
3'0
40
50
60
Duration in Days
!FIGURE 11.3!
Duration ofEconomic Events
277
Hazard
1.61.51.4l.3-
1.2I.I 1.00.90.80.70.60.50.40.30.20.10.00
I
IO
20
Jb
sh
40
Duration in Days
I
60
I
80
!FIGURE 11.4!
11.2.5 Parametric Estimation
As developed in Chapter 8, the parameters of a distribution can be estimated
by maximum likelihood. Ifwe have full data, then the likelihood function is
n
L(O) = Ilf(tj ;O),
J =I
(11.15)
where O is a vector of one or more parameters and/(tj ;O) denotes the PDF at
the /h duration. When the data are censored, as with the strike data, the
contribution to likelihood of a censored observation is the SDF evaluated at
the censored duration, which is 80 in this case. We define the indicator
variable Cj to be cj = I if the end of the strike is observed (i.e., not censored)
and bj = 0 if the/h observation has been censored. Then the likelihood is
n
L(O) = IT [f(tj ;0)]6j · [S(9;0)] 1 -<\
J=I
(11.16)
Applications in Economics and Finance
278
and the log-likelihood is
n
n
£(0) = L 61 · In [f(tJ;O)] + L(l-61 ) · In [S(t1;0) ].
J=I
J=I
(l l.16a)
Substituting/(t1;0) = S(lj;O) · >.(lj;O), we have
£(0) =
L=I In [S(t ;0)] + L=I 6 · In [>.(lj;O)] .
n
n
1
J
( l 1.16b)
1
J
If we select the exponential distribution as our model for the strike
data, we have the well-known result
n
I\
I\
0 - >.
"£,61
J=I
(11.17)
n
;:, lJ
J=I
n
as derived in Chapter 8 (see Section 8.2), where "£, 81 is the number of ob­
J=I
n
served strike endings and "£, t1 is the exact exposure of being on sttike.
J=l
62
62
From the data in Table 11. l we calculate "£, 9 = 2118 and "£, 61 = 50,
J=I
J=l
f�8 = .0236.
giving us� = 2
IJEXAMPLE 1ij]j (a) Find the likelihood equations obtained if a Weibull
distribution is fit to the strike data. (b) Verify that the MLE's of the Weibull
I\
I\
parameters are a= .0325 and {3 = .9210.
!SOLUTION!
(a) From Equation (11.1) we have >.(t) = a/3 · 1!3 -I and from Equation
13
(11.2) we have S(t) = e-o:·1 • Then the log-likelihood from Equation
( l l.16b) is
Duration ofEconomic Events
279
62
62
f(a,,8) = I:1n(e-a·f) + L6J · ln(a{J
J=I
J=I
-1-1)
Q
Q
Q
Q
J= l
J =l
J=I
J=I
= -aI:t +ln aI:o1+ ln,BI_>51+(,8-l )L6rln9.
62
62
From Table 11.l we have I:61 = 50 and I:61 · ln 9 = 134.6239, so our
J=I
J=I
log-likelihood simplifies to
f(a,{3) =
Then
62
- aL if+50 ·In a+50 · ln ,8+134.6239(,8-1).
J= I
62
of(a,,8)
= -I:1+�=0
oa
J=l
and
(l l .18a)
"'2:/J · In 9 + 5j + 134.6239 = 0 (l l .18b)
62
of(a,,8)
-a
8,8 =
J=I
are the likelihood equations that must be solved by an iterative process.
(b) Let (3 = .9210. From Table 11.1 we find the values
A
1·
62 "
E1
= 1536.8773
I
Q "
J=
and Y:,
In t1 = 5995.0653. Substituting these values, along with <i =
J=I
.0325 and (3 = .9210 into Equations (l l.18a) and (l l .18b), we obtain
and
oR.(a,/3)
- l536.8773 + .0��5 = 1.5842
oa =
af(a,(3) =
o{J
- (.0325)(5995.0653) + _925010 + 134.6239 - 5.9269,
where the results differ slightly from zero due to rounding.
0
Applications in Economics and Finance
280
Recall that when /3 = 1, the Weibull distribution reduces to the exA
A
ponential. Here our value of /3 = .9210 is close enough to f3 = l to justify
our choice of the exponential distribution in the first place.
11.2.6 Use of Concomitant Variables
Intuitively, the distribution of strike durations may vary according to various
characteristics of the labor group involved, such as ( 1) size of group, (2) type
of industry, (3) time of year, (4) disputed issue(s), and (5) current state of the
economy. As introduced in Section 8.5, these concomitant variables can be
represented in a parametric model fit to the data. In the case of the strike
. data, the first four of the influences listed above were taken into account by
considering only data for which these variables were constant: l ooo+ size
group, manufacturing industry, June strikes, and wage disputes.
In this section we expand our parametric model to include a concom­
itant variable addressing the general economic condition, an index of indus­
trial production. We choose the Cox model given by Equation (8.77), with
constant underlying hazard ,\ = e00 , and concomitant variable z1 represent­
ing a fixed value of the index of industrial production associated with each
strike. Then the hazard function is
(1 l. 19)
and the survival function is
(11.20)
so the log-likelihood function, from Equation (I l.16b), is
e(ao,a1) = -
:z:= 9. e + = + I: oj(ao+a1z1).
n
j =I
0 0 01 1
n
j=I
(11.2 l}
When the values of z1 are assigned to each strike (see Kennan [45]), then
Equation (I l .21) can be solved for the MLE's of ao and a1, obtaining the
values ao = -3.72 and al = 10.21.
When industrial production is on its trend line, so that z1 = 0, then
the constant hazard function would be ,\ = e00 = .02423, and the expected
strike length would be ± = e-00 = 41.26.
Mortgage Loan.Prepayments
281
11.3 MORTGAGE LOAN PREPAYMENTS
In this section we will briefly explore the survival pattern of amortized loans,
specifically those secured by a mortgage on real property. We assume that
the actuarial reader is familiar with the mathematics of such loans, as des­
cribed, for example, in Kellison [44].
When we speak of the survival pattern of an entity, there must be
some definition of what it is that constitutes survival and, consequently,
what constitutes lack of survival, or failure, of the entity. Consider a con­
ventional mortgage loan of amount L, to be repaid over 30 years by 360 level
monthly payments. If we were to define "survival" to be the continuation of
the regular monthly payments, and "failure" to be the complete repayment of
the loan, then the observ.ed survival pattern, measured in months, for a loan
that made all regular payments would be S0 (t) = 1.00 for O :::; t < 360, and
S0(360) = 0.
Most mortgage loans, however, do not behave with this pattern of
perfect regularity, since there are two significant events that could occur to
disturb it, namely default and prepayment. These two risks are discussed in
Section 11.3.1. The major use of mortgage loan survival patterns is in
conjunction with mortgage-backed securities (MBS), so an introduction to
this investment instrument is presented in Section 11.3.2. Finally, in Section
11.3.3 we describe various models of mortgage loan survival in light of the
default and prepayment risks that have, or have had, some use in the invest­
ment arena.
I I.3.1 Default and Prepayment Risks
Mortgage default occurs if the borrower discontinues making the regular
monthly payments. The probability of default varies with considerations
such as the loan-to-value ratio, a measure of the borrower's equity in the
property, the payment-to-income ratio, a measure of the borrower's finan­
cial ability to make the regular monthly payments, and whether or not a
second mortgage exists. Experience data shows that the default rate, as a
function of the duration of the loan, reaches a maximum at about 2-4 years
from origination, and then declines with duration. If the mortgage is in­
sured, however, then default simply results in the immediate payment of the
outstanding balance of the loan to the lender by the insurer. Mortgage
insurance is available from government agencies, such as the Federal
Housing Administration (FHA) or the Veterans Administration (VA), and
from private mortgage insurance companies.
282
Applications in Economics and Finance
Since, when the mortgage is insured, default can be viewed as mere­
ly one of the reasons for prepayment, we need consider it no further for our
purposes in this text.
Because of the prepayment option, the right granted to the borrower
to prepay the loan, an investor in a mortgage cannot be certain of the future
cash flow. An intended 30-year investment could tum out to last the entire
30 years, or only a year. This uncertainty about the future cash flow due to
the prepayment option constitutes the prepayment risk.
Prepayments occur for various reasons, including the following:
(I) Sale of the property by the owner, due to geographic relocation,
trading up, or property settlement upon divorce.
(2) Repossession and sale of the property upon default by the bor­
rower.
(3) Receipt of insurance proceeds in the case of a property de­
stroyed by fire or other insured event.
(4) Mortgage refinancing as a result of falling interest rates
In addition to the above reasons for the total prepayment of the outstanding
balance of a mortgage loan, borrowers will frequently make additional pay·
ments of principal beyond the amount included in a particular monthly pay�
ment. Such partial prepayments are called curtailments.
In order to analyze the quality of a mortgage, or a pool of mortgages,
as an investment, and to determine the appropriate price to pay for such an
investment, a projection of prepayments, or a prepayment model, is needed.
We will discuss such models in Section 11.3 .3.
11.3.2 Overview ofMortgage-Backed Securities
The original lender under a mortgage loan, called the mortgage originator,
wil I tend to be an institution such as a commercial bank, thrift, mortgage
banker, or pension fund. In order to make investment in mortgages available
to smaller investors, and to increase the total supply of capital available for
such investments, a substantial secondary mortgage market has been devel­
oped. 2 The purchases by secondary investors of entire loans from mortgage
originators is called the secondary whole-loan market, and it has been in
operation for many years. However, the cost, paperwork, and lack of a large
and active whole-loan market (lack of liquidity) prevents many small inveszsee, for example, Fabozzi [27].
Mortgage Loan Prepayments
283
tors from entering this market. To correct this, the concept of the mortgage
pass-through was created.
In this arrangement, a business entity purchases a large pool of mort­
gages, and then sells securities, called mortgage-backed securities (MBS),
representing shares in the cash flow generated by the mortgages. The basic
type of MBS are called mortgage pass-through securities, or simply pass­
throughs.3 The mortgages backing the MBS can be residential or com­
mercial, and can be guaranteed by an agency of the government (agency
mortgage-backed securities) or not (nonagency mortgage-backed securi­
ties). The major players in the agency MBS field are the Federal National
Mortgage Association (FNMA), affectionately known as "Fannie Mae," the
Government National Mortgage Association (GNMA), or "Ginnie Mae," and
the Federal Home Loan Mortgage Corporation (FHLMC), or "Freddie Mac."
The borrowers make their regular monthly interest and principal
payments, as well as unscheduled prepayments of principal. After the deduc­
tion of a service fee, the cash flows from the mortgage payments are "passed
through" to the holders of the securities in proportion to their shares of
ownership of the pool. In addition to making investment in mortgages avail­
able to a broader range of investors, the pooling and pass-through arrange­
ment pools the prepayment risk from individual mortgages, making prepay­
ment prediction more reliable. This concept of risk-pooling is a fundamental
one in actuarial science.
11.3.3 Survey ofMortgage Prepayment Models
It is easy to see that a major factor involved in evaluating and pricing an
MSB is the projected prepayment rate associated with the pool of mortgages.
Suppose a pool of 100 30-year mortgages, each for an original
amount of $100,000, is established. The initial outstanding balance of size
Bo= 10,000,000 then declines to zero over the next 360 months as payments
are made. This initial group of dollars of outstanding balance is analogous
to an initial group of new-born lives, denoted f0, and constitutes the radix of
a life table showing the surviving dollars of outstanding balance at each
subsequent duration n, denoted Bn , where n is measured in months. Sup:­
pose, for the moment, that no prepayments are made. Then the graph of Bn ,
shown in Figure 11.5, declines slowly at first, since most of the regular
monthly payment is interest, and then more rapidly as the principal repay­
ment portion of the monthly payment increases.
3Two interesting variations of this are collaterali;,ed mortgage obligations and
stripped mortgage-backed securities. See, for example, Chapter 26 of Fabozzi [27].
Applications in Economics and Finance
284
Outstanding Balance
(in millions of dollars)
10
8
6
4
2
0
60
120
180
240
300
360
Duration in Months
!FIGURE 11.S!
When prepayments are assumed in the model, the graph of outstanding
balance will decline more rapidly. In the remainder of this section we will
describe five historical and currently popular survival models of the out­
standing balance (hence models of the prepayment pattern) under mortgage
loans.
Notationally, we reserve the symbol Bn for the outstanding balance
at duration n in the absence of prepayment, and use Cn to denote the same
concept in the presence of prepayment.
Early FHA data showed that the average duration at which mort­
gages terminated due to prepayment was about 12 years. This fact lead to
the adoption of the twelve-year prepaid life model, under which it was
assumed that there were no prepayments at all for the first 12 years of the
mo1tgage, followed by full prepayment at duration 12, or 144 months.
Under this model we would have Cn = Bn for O ::::; n < 144, and Cn = 0 for
n � 144. The graph of Cn under this model, and its relation to the graph of
Bn , are shown in Figure 11.6.
The twelve-year prepaid life model is rarely used in practice today,
in light of its readily apparent shortcomings.
Mortgage Loan Prepayments
285
Outstanding Balance
(in millions of dollars)
10
8
6
4
2
0
60
120
180
240
300
360
Duration in Months
!FIGURE 11.61
A method commonly used today is the constant prepayment rate
model (CPR). Under this method, the amount of principal prepaid in the n'h
month, denoted PPn , is a constant percentage, denoted SMM, of the out­
standing principal balance at. the beginning of that month reduced by the
regular (scheduled) principal repayment. (Note the important distinction be­
tween scheduled principal repayment and unscheduled principal prepay­
ment.) Then we have
(11.22)
where PRn denotes the scheduled principal repayment.
Next we define
(11.23)
to be the proportion of the outstanding balance that would have existed at
duration n in absence of prepayment, denoted Bn , ,that actually does exist at
duration n in light ofthe prepayments, denoted Cn. Ifwe think ofthe pool as
starting with 10,000,000 unit mortgages of $1 each, then Qn represents the
percentage of them still in effect (i.e., not prepaid) at duration n, and 1-Qn
represents the percentage ofthe initial pool already prepaid by duration n.
286
Applications in Economics and Finance
!I EXAMPLE l l.6!j Show that SMM, as defined by Equation ( 11.22), can be
written as
SMM = 1- Qn
Qn -I'
(11.24)
where Qn is defined by Equation (11.23).
!SOLUTION! First note that the total reduction in outstanding balance at
duration n, which is given by Cn - I - Cn , is made up of the scheduled
principal repayment plus the unscheduled principal prepayment, leading to
PPn = Cn -1 -Cn - PRn . In turn, PRn would have been Bn-1 - Bn if the
outstanding principal at duration n - I were Bn-I· Since it is Cn -1 instead,
then we have the proportional amount
= Cn-1 - BnB·nC-1n -1
Then from Equation (11.22) we have
SMM=
Cn -t -PRn
Cn -t -- Cn - PRn
Cn-1 - PRn
lI-
,
Bn·Cn-1
Cn -1 - [Cn -1 -- ---· ]
Bn-t
Cn/Bn
C11-1/B11-I
= I - _ih__
Qn-1
as required.
D
We observe that SMM, which is called the single montltly mortality
for the mortgage pool, is in the nature of an effective monthly rate of dis­
count. On the other hand, the constant prepayment rate (CPR) for which this
Mortgage Loan Prepayments
287
method is named, is generally expressed as an annual rate. Then analogous
to the familiar compound interest formula
d=l-
(
cf-12)) 12
1-12
(11.25)
we have here
CPR = l - (l-SMM) 12 .
(l 1.26)
The graph of Cn under this method, and its relation to the graph of Bn , are
shown in Figure l l.7 for SMM = .Ol.
Outstanding Balance
(in millions of dollars)
300
360
Duration in Months
!FIGURE 11.7!
A third prepayment model, once widely used but less common to­
day, is one derived from experience under 30-year FHA-insured mortgages.
The FHA model is a true survival model, estimated from sample data, show­
ing values of S(t), 0 $ t $ 30, the probability of surviving to each annual
duration t, as well as values of q 1, 0 $ t $ 29, the percentage of mortgages
expected to terminate in each mortgage year. Then a particular mortgage
pool can express its prepayment model as a percentage of the FHA experi­
ence model.
288
Applications in Economics and Finance
The advantage of the FHA model over the CPR model is that it
better reflects the effect of mortgage duration on prepayment. Its major
disadvantages are that many users find it to be complex to use, as well as
the adjustments required when the FHA data is updated.
The FHA model clearly shows that prepayment rates are lower in the
early years of a mortgage and then increase with duration, whereas the CPR
model uses the same prepayment rate throughout. Our fourth method, the
Public Securities Association model (PSA), currently viewed as the industry
convention, combines the information contained in the FHA data with the
simplicity of the CPR approach. Under this method, the CPR is assumed to
be .002 ( or .2%) in the first month, increasing by .002 per month to a level of
.06 at duration 30 months, and remaining constant at that level. Then a
particular mortgage pool can express its prepayment assumption as a per­
centage of the PSA model.
Graphs of Cn , the outstanding balance of a mortgage pool, using
various multiples of the PSA prepayment. model are shown in Figure 11.8.
Note that Bn is the special case that results from assuming 0% of PSA as the
prepayment model
Outstanding Balance
(in millions of dollars)
B = 0%PSA
� n
8
-----50%PSA
6
----120%PSA
4
200%PSA
2
0
60
120
180
240
300
360
Duration in Months
!FIGURE 11.81
A fifth approach, the econometric prepayment model, projects
SMM values for each remaining month of the MBS, taking into account such
Mortgage Loan Prepayments
289
economic factors as the current housing turnover and refinancing rates as
well as the age and seasonality variables of the mortgage pool. The advan­
tage of econometric models is that they incorporate changing economic fac­
tors which may have made recent experience data an unreliable predictor of
future prepayments. Their major disadvantage is again found in their com­
plexity. For practical use an econometric model is frequently expressed as
an equivalent average CPR or percentage of PSA.
11.4 CORPORATE BOND DEFAULT RATES
In this section we will briefly examine the topic of measuring the rate of
default of corporate bonds. The actuarial reader is assumed to have a basic
knowledge of the nature of corporate bonds, as described, for example, in
Chapter 7 of Kellison [44].
Modern empirical measurement of the bond default risk was pio­
neered by Hickman [35] in 1958, and has been an area of active analytic
interest. The material in this section of this text is based on Altman [2].4
11.4.1 Historical Approaches to Measuring Default Rates
The accurate measurement of the default risk is critical in the pricing of debt,
measuring bond performance, and assessing bond market efficiency.
Early measurement approaches calculated the default rate in a par­
ticular year of a bond's term by dividing the value of defaults by the original
value of the bond issue, rather than by the surviving population of bonds at
time of default. (The actuarial reader will recognize the former measure as
an unconditional one, analogous to the life table concept denoted by n lqo,
and the latter measure as a conditional one, analogous to the life table con­
cept denoted by qn .) The former approach fails to consider that a bond can
"die" in ways other than through default, such as by call redemption, sink­
ing fund, or maturity. These annual default rates were then averaged over
several years to obtain an average annual rate. For example, for the period
1978-87, the average annual default rate, measured in the traditional way,
was 1.86% per year.
The traditional default rate measurement uses the par (or other)
value of the defaulting bond in the numerator of the rate calculation. A more
4In the abstract to his paper, Altman observes that his approach seeks to measure
bond "mortality" in a manner similar to the way actuaries assess the mortality of human
beings.
29 0
Applications in Economics and Finance
relevant measure for investors is not the rate of default, per se, but rather the
rate of investment value actually lost (default loss rate measurement).
Suppose an investor purchases a bond at par value. The bond defaults by
failing to make a particular coupon payment, and the investor sells the bond
immediately after the defaulted coupon date at 40% of its par value. Then
the default loss rate calculation would use a loss of 60% of par plus one
coupon in the numerator, whereas the traditional default rate calculation
would use the entire par value. By taking into account that the defaulting
bond can be sold, on the average, for about 40% of par, the 1978-87 average
annual default loss rate was about 1.20% per year, compared to the l .86%
traditionally measured default rate.
!!EXAMPLE 11.711 On January 1, 1990, there were 1,000,000 bonds of
various issuers (but of similar quality) purchased by a pension fund in units
of $ I 00 face amount each. Each bond paid l 0% annual coupons on
December 31, was purchased at par value, and had a maturity date of Dec­
ember 31, I 994. In all cases the bonds could be redeemed early on
December 31, 1992 or December 31, 1993. In each of 1990, 1991, and 1992,
l 0,000 bonds defaulted and were disposed of by the investor. On December
31, 1992, 100,000 bonds were redeemed. In I 993, 20,000 bonds defaulted
by not paying the 12/31/93 coupon, and were sold for 40% of par value by
the investor. Calculate the 1993 default rate (a) as an unconditional measure,
(b) as a conditional measure of default rate, and (c) as a conditional measure
of default loss rate.
ISOLUTION!
(a) The par value defaulting in 1993 is 2,000,000 out of an original issue of
100,000,000. The unconditional default rate is therefore
2, 000,000
100'000'000 = ·02000
(b) For the conditional measure, the exposure base must reflect the 130,000
bonds (of 13, 000,000 par value) that were redeemed or disposed of due
to default in prior years. Then the traditional measure of default rate is
2,000,000
87'000'000 = . 02299
(c) The annual coupon due 12/31/93 is $1 0 per bond, so 200,000 is lost in
defaulted coupons. By selling the defaulted bonds at 40% of par, then
Corporate Bond Default Rates
291
(.60)(1 00)(20,000) = l,200,000 is the amount lost due to default m
addition to the lost coupons. Then the conditional measure of default
loss rate is
l,400, 000 _ 0 1609
D
·
87,000,000 ·
11.4.2 Altman 's Bond Mortality Rate Concept
As suggested in part (c) of Example 11.7, the "actuarial method" advocated
by Altman tracks the survival pattern of a cohort group of bonds over time.
In his model, bonds can exit from the original population by either default,
calls, sinking funds, or maturities, so that we have a multiple-decrement
environment. (To measure default rates, the non-default exits can be com:..
bined so that the model becomes double-decrement only.) Then if D(t)
denotes the value of defaulting debt in year t and P(t) denotes the value of
the surviving population of bonds at the start of year t, then
MMR, = !21!)_
P(t)
(11.27)
gives the marginal mortality rate for year t.
Altman further defines
SR, = 1 - M1'..fR1
(11.28)
to be the bond survival rate in year t, and
CMR, = 1 - Il SR.i
j=l
I
(11.29)
to be the cumulative mortality rate over the interval (0, t).
Finally, Altman considers the mortality/survival experience of vari­
ous classes of bonds over the seventeen-year sample period of 1971 through
1987. Since the values of MMR, were calculated at each duration t for each
separate issue year, then they must be combined to reach a rate for duration t
for the entire study. The combining is done on a value-weighted basis, since
an unweighted average could be misleading if new issue amounts varied
significantly from year to year. The reader interested in Altman's numerical
results is referred to [2].
Applications in Economics and Finance
292
11.5 EXERCISES
11.1 Introduction;
11.2 Duration of Economic Events
11-1. Let T have the Weibull distribution given by Equations (l l.l) to
(11.3). Consider the transformation of duration given by s = tf3.
(a) Using Equation (2.65), show that.fs(s) is the PDF of an exponen­
tial distribution.
(b) Using Equation (2.66), show that >.s(s) is the HRF of an exponen­
tial distribution.
Exercises 11-2 through 11-5 relate to the log-logistic distribution defined by
Equation (11.8).
11-2. Derive an expression for the SDF, S(t).
11-3. Derive an expression for the PDF, f(t).
l l'-4. Show that the distribution has negative duration dependence if
O<o�l.
11-5. If o > l, find the value oft at which the duration dependence changes
from positive to negative.
11-6. Using the strike data presented in Table 11.1, estimate S(20) by the
/\
Nelson-Aalen method. Compare the result with the value of S(20)
shown in Table 11.1.
11-7. Repeat the estimation of the exponential distribution parameter for
the strike data if the censored data are treated as complete.
11-8. The variance of the MLE of the exponential distribution parameter
/\
.
.e(>.) }-1
can be approximated by { - ;:>2a>.2
• Estimate the variance of A
for the strike data.
11.3
Mortgage Loan Prepayments
11-9. Show that Cn = Bn · (1-SMMt, where Bn , Cn , and SMM are
defined in Section 11.3 .3.
Exercises
293
11-10. Find the value of SMM at durations beyond 30 months in the PSA
model.
11-11. Find the value of SMM in the 201h month under a 150% of PSA
model.
11-12. A pool of 30-year mortgages with initial outstanding balance of
Co = I 00,000 is being amortized at a nominal annual interest rate,
convertible monthly, of ;<12> = .12. The pool is subject to prepayment
at the constant rate of CPR= .05838. Find the values of Cn , for
n = 1,2,3,4,5.
11-13. Repeat Exercise 11-12 using a prepayment model of I 00% of PSA.
11.4
Corporate Bond Default Rates
11-14. What is the standard actuarial symbol for each of the bond mortality
symbols defined by Equations (11.27), (11.28), and (11.29)?
11-15. In Altman's study of bond mortality over the period 1971-1987, the
total new issue amount of B-rated debt was 43,227 (in millions of
dollars). Assume there were no early redemptions of B-rated bonds.
The following table shows the dollar amount of default (in millions)
on such bonds by duration.
Years
after
Issue
1
2
3
4
5
Dollar
Amount
of Default
856
390
311
1767
1660
6
7
8
9
10
1905
1315
1411
2847
1332
Find the values of each of the following.
(a) MRR4
(b) SR1
(c) CMR10
CHAPTER12
ADDITIONAL MISCELLANEOUS APPLICATIONS
12.1 INTRODUCTION
In this final chapter we present a number of miscellaneous applications of
the basic survival model estimation theory and techniques developed earlier
in the text. An attempt is made to draw these applications from as wide a
range of disciplines as possible.
The applications are organized primarily into the two major groups
of parametric models, with emphasis on alternatives to the convenient expo­
nential distribution (Section 12.2), and nonparametric models (Section 12.3).
Within each section the examples are further divided as to specific type of
model and estimation technique. Both complete and incomplete data samples
will be encountered in the estimation work.
12.2 PARAMETRIC ALTERNATIVES TO EXPONENTIAL MODELS
As we have seen several times in earlier chapters, the exponential distribu­
tion is frequently chosen to model the failure time random variable, due, in
part, to the convenience provided by its constant hazard function. It is not
reasonable, however, to expect that the constant hazard rate property applies
to very many natural survival patterns. In this section we present several
examples of survival model estimation using hazard rates that are not con­
stant functions over the entire domain of the primary variable.
12.2.1 Piecewise-Exponential Models
A simple extension of the exponential model is to assume that the hazard
rate remains constant at level >. 1 up to time (or age) y, and is then constant at
level >.2 thereafter. Then the piecewise-exponential model is defined by its
HRFas
0 � t <y
t z. y
( 12.1)
Additional Miscellaneous Applications
296
The survival function is
S(I) -
{
0:St<y
::::•. e-c,-,i>,
t ?_y
{.
and the density function is
f(t) =
q
)q e-t·>
O:St<y
>.2 . e-Y·-'1 . e-(1-y),\z
t?:. y
(12.2)
(12.3)
!!EXAMPLE 12.l!j Use maximum likelihood to estimate the parameters of
the piecewise-exponential model given by ( 12.1 ), using the following data.
Unit
A
B
c
D
Outcome
Failed
Withdrew
Failed
Withdrew
Time of Outcome
(A
<
ts y
<y
tc
>y
ID> Y
!SOLUTION! From Chapter 8 we know that the contribution to likelihood
of the i1h failure is the PDF f(t;), and the contribution ofthe}'h withdrawal is
the SDF S(tj). Thus we have
L()q, .\2 ) = () q . e-1A·A1 )(e-te·A, )(,\ 2 . e-y·>-1 • e-<rc-y)A2)(e-y·.J\1 . e-<10-y) .\2 ).
The log-likelihood is
where E; denotes the exact exposure to hazard .\;. Then
8£( .\..h..,\2)
and
a>,i
= -1. - E1 = 0
)q
at(>q, >.2) = J_ -E
8X 2
\2
z
0
Parametric Alternatives to Exponential Models
lead to the MLE's �1 =
fi
Number of failures
ar orm Exact exposure
297
i, and �2 = L, which we note to be of the familiO
Example 12.l is easily extended to the general case of n subinter­
vals. If the ;th subinterval has constant hazard rate A;, and if a sample
produces d; failures with subinterval i with exact exposure E;, then the MLE
of A; is the familiar
A;
(12.4)
!!EXAMPLE 12.2[ Let D; denote the (random) number of failures in a time
interval for which the hazard rate is constant at A; and let d; denote the
sample realization of D;. Let the exact exposure of the sample within the in­
terval be E;. Conditional on E;, let D; have a Poisson distribution with mean
E[D; IE;] = A; E;.
d.
Show that the MLE of A; is £;
!SOLUTION! Recall that the Poisson probability function is given by
x
· �-m, where m = E[X]. Then in this case the likelihood is
Pr[X = x] = m x.
L(,\;) =
(>. El· e -H·
; ; d;; ' ' and the log-likelihood is
£(;\;)
= d; ·In(;\;£;) - A; E; - Ind;!
- d; · In;\; - .X; E; + C,
where C is a constant not involving A;. Then, clearly,
d· as require
• d.
so that A; = i;,
A
0
�J:;) = f; � E; = 0,
O
12.2.2 General Composite Hazard Rate Models
The model of Section 12.2.1 is a simple example of a composite hazard rate
model, where each segment in the composite model has a constant hazard
rate. This simple case is illustrated in the following figure, where we as­
sume that the hazard increases with duration or age.
298
Additional Miscellaneous Applications
,\(t)
!FIGURE 12.1!
Note that Figure 12.1 assumes that the hazard remains constant at ,\(t) = -\5
for all t 2: y4.
In this section we consider more general cases of the composite
hazard rate model. In reliability analysis of mechanical devices, for example,
the hazard rate function frequently has the "bathtub" shape illustrated in the
following figure.
,\(t)
0
!FIGURE 12.21
Parametric Alternatives to Exponential Models
299
First there are early failures resulting from material or manufacturing de­
fects. As this early period passes, the failure rate declines rapidly to the flat
portion of the hazard curve where the rate is relatively constant, representing
chance failures caused by sudden stress, unpredictable operating conditions,
and so on. Finally, as the devices age, the hazard rate increases due to equip­
ment deterioration, accumulated shocks, and so on.
With modem quality control testing, however, many initial product
failures can be eliminated so that the initial (decreasing) portion is now a
less common feature of the hazard pattern. Instead we can now expect to
observe a nearly constant hazard rate for some initial period of time, after
which it strictly increases from its original value. For example, the model
could be piecewise linear, as shown in Figure 12.3a, or it could be composed
of a constant portion followed by a quadratic portion, as shown in Figure
12.3b.
!FIGURE 12.Jh]
!FIGURE 12.3a!
The piecewise linear model of Figure l 2.3a has been explored by Shooman
[70]. The constant-and-quadratic combination of Figure 12.3b, which we
will describe in the remainder of this section, is discussed by Ananda and
Singh [3].
Let the composite hazard rate model be defined by
,\(t)
c
0 :St <y
+ /l(t-y)'
t '?. y
(12.5)
where .\, (3, and y are unknown parameters. The corresponding survival
function for O :S t < y is the familiar S1 (t) = e->.,. Fort "?. y, we have
Additional Miscellaneous Applications
300
S2(t)
1
[
-1
·
=
· [- =
= [- -
(.\+,8(r-y)2 )dr]
e-y>.. exp
e-y>.. exp
A(t-y)
exp
q(t-y)J ],
Al
q(t-y)3 ]
so the composite survival function is
S(t) =
{
exp[
-
.\t]
exp[-At
-
o::;1<y
q(t-y)3 ]
.
(12.6)
t?:.y
In estimating the unknown parameters, we will make use of the cumulative
hazard function (CHF), given by
A(t) =
- In S(t) =
{
At
c
o::;,<y
,\t + q(t-y)3
I"?_ y
.
(12.7a)
By defining the indicator variable l(t) to be
l(t) =
o::;t<y
t "?. y
,
then (12.7a) can be written as
A(t) = ,\t + q(t-y)3 • I(t) .
(12.7b)
The data from which the model parameters will be estimated are
taken from Barlow, et al. [7]. A sample of n = 49 Kelvar Epoxy Spherical
pressure vessels were subjected to constant sustained pressure at the 70%
stress level until all had failed, so that we have complete data with exact
times of failure. The failure times (in hours) are shown in the following
table.
301
Parametric Alternatives to Exponential Models
1051
1337
1389
1921
1942
2322
3629
4006
4012
4063
4921
5445
5620
5817
5905
5956
6068
6121
6473
7501
7886
TABLE 12.1
8108 10205
8546 10396
8666 10861
8831 11026
9106 11214
9711 11362
9806 11604
11608
11745
11762
11895
12044
13520
13670
14110
14496
15395
16179
17092
17568
17568
To estimate our model parameters, we first construct the empirical
(observed) survival function S0(t), defined by Equation (4.1 ), and the
empirical CHF by the relationship A0 (t) = - lnS0 (t). Selected values of
S0 (t) and A0 (t) are shown in Table 12.2.
TABLE 12.2
t
S0 (t)
1000 1.000
2000
.898
3000 .876
.857
4000
.776
5000
A0 (t)
0.000
.108
.131
.154
.254
6000
7000
8000
9000
10000
.673
.612
.571
.490
.429
.395
.491
.560
.714
.847
11000
12000
13000
14000
15000
.367
.204
.184
.143
.102
1.001
1.589
1.695
1.946
2.282
16000
17000
18000
.082 2.506
.061 2.793
.000
00
Additional Miscellaneous Applications
302
The empirical CHF, A0 (t), is illustrated in the following figure.
AD (t)
3.00
2.80
2.60
2.40
2.20
2.00
1.80
1.60
1.40
1.20
1.00
.80
.60
.40
.20
0 1 2 3 4 5
6 7 8 9 10 11 12 13 14 15 16 17 18
(in thousands)
!FIGURE 1M}
Parametric Alternatives to Exponential Models
303
Using a minimum area criteria (see Ananda and Singh [3]), the opti­
mum value of y is formed to bey = 3400. Then taking this optimum value
as the estimate of y, the least-squares estimates of .X and /3 can be obtained
from the regression model
c = .XX1 + /3x2,
(12.8)
where c = A0(t), x 1 = t, and x2 = -!(t-y)3 · l(t). For this data, the results
are A = .0490 and /3 = .0038. The fit of the data to the model can be
analyzed by a chi-squared test by first grouping the data into intervals and
calculating the expected number of failures in each interval according to the
model. These results are shown in Table 12.3. The p-value associated with
this test is .3296.
/\
/\
Failure Time Interval
(in thousands)
0.0 - 1.5
1.5 - 3.5
3.5 - 5.5
5.5 - 7.5
7.5 - 9.5
9.5 - 11.5
11.5 - 13.5
13.5 - 15.5
15.5 and up
TABLE 12.3
Observed Number Expected Number
of Failures
of Failures
3
3.53
3
4.31
6
4.34
7
5.94
7
8.04
8
8.83
6
7.30
5
4.34
4
2.37
The overall results show that the constant-and-quadratic composite hazard
rate function is an appropriate model to describe this failure time data.
12.2.3 Polynomial Hazard Rate Models
Models with a hazard rate function given by a low degree polynomial (usual­
ly not greater than 3) are very commonly used to model the distribution of
survival times in clinical studies. The simplest member of this family, of
course, is the exponential distribution, whose constant hazard rate is a poly­
nomial of degree zero.
The next member of the polynomial HRF group is the Rayleigh distri­
bution, for which the hazard rate is a linear function. We define the Rayleigh
distribution by its HRF, SDF, and PDF as
304
Additional Miscellaneous Applications
(l2.9a)
and
2
S(t) = exp[ - (,\0t+,\1 t )],
(12.9b)
f(t) = (,\o+2,\1t) · exp[ - (>.0 t+,\ 1 t 2 )],
(12.9c)
respectively, where ,\0 > 0 and ,\ 1 > 0.
If we have full data with exact times of failure, then the Rayleigh
parameters can be estimated by maximum likelihood, where each failure
contributes its PDF to the likelihood. Thus if there are n members in the
sample with failure times t1, t2, ... , tn, the likelihood is
n
t, -,\ t ,,'),
L(>-.o, .\1) = TI { (>.o+2,\1t;) · exp[ - (>.ot;+A1t/)]}
i=I
,!'.\< Ao+2,\
and the log-likelihood is
1 ,, ) •
exp ( - Ao
1
n
n
n
i=I
i=l
i=I
£(>.o,>.1) = Lln (>.o+2>.1t;)-,\0Lt;->.1Ltf.
(12.I 0)
(12.11)
The likelihood equations, to be satisfied by the MLE's of ,\0 and >. 1 , are
and
n
n
i=I
i=l
n
n
i=l
i=l
L(>-0+2,\,t;)-1 - I:!; = o
2Lt;(Ao+2.\1t;r 1 -I:tf = 0.
{12.12a)
(l2.12b)
If we multiply (12.12a) by >.0 and (12.12b) by ) q , and add the results, we
obtain
n
Ao -
n - A1 Et;
n
i=I
'£,t;
(12.13)
i=I
Then substituting(12.13)for ,\0 in (12.12a) we obtain
( 12.14)
Parametric Alternatives to Exponential Models
305
where S1 = I:t; and S2 = I:t;. Finally, Equation (12.14) is solved for .\1
11
i=l
n
A
i=I
by iteration and .\o is obtained by substituting this value of.\ 1 into Equation
(12.13). This procedure is illustrated in the following example.
A
A
!!EXAMPLE 12.31[ The times until remission (in weeks) for 21 leukemia
patients under a particular treatment, taken from Gehan [30], are given in
Table 12.4.
TABLE 12.4
Time until Number of
Remission
Patients
l
2
2
2
3
I
4
2
5
2
8
4
2
11
12
2
15
I
I
17
22
I
23
l
Estimate the parameters of a Rayleigh distribution fit to this data by maxi­
mum likelihood.
!SOLUTION! We haven= 21, S1 = 2::t; = 182, and S2 = '2:.,t/ = 2414.
Then (12.14) becomes
21
21
i=I
i=I
L 21 - 24141 + 364.\ 1; - 1 = o.
21
i=l
1
1
Using the Newton-Raphson iterative procedure with initial estimate taken to
A
be .\ 1 = n = .0087, the successive values are .0074, .0059, .0046, .004O,
s
2
and then converging to .0039 (within four decimals) as our estimate )q. Sub.
.
\' = 21 -(.0039X2414) = .0637 · D
. . th.1s mto
. (12 . 13) gives
st1tutmg
"0
E quat1on
A
182
Additional Miscellaneous Applications
306
If the observations are censored after the first d failures, then s = n-d
do not fail and the likelihood function (and the resulting likelihood equa­
tions) are modified. This situation, using the data of Example 12.3, is pur­
sued in the exercises.
Next we consider an example of a quadratic hazard rate model using
grouped data, with parameters estimated by least squares. The data and pro­
cedure are taken from Lawless [50].
Let the HRF be given by
(12. l5a)
so that the CHF is
(12.lSb)
and the SDF is
(12. l5c)
Let the data be grouped into k+ 1 intervals (not necessarily of equal length),
with no censoring of observations except perhaps in the final interval, as
illustrated in the following figure.
0
2
l
k-1
3
k
!FIGURE 12.5!
If the sample is of size n and the observed number surviving to duration t, for
t = 1, 2, ... , k, is denoted by n,, then S 0 (t) = !Ji, the observed proportion
surviving to duration t, can be taken as a preliminary estimate of S(t), and
A0(t) = -lnS 0 (t) is a preliminary estimate of A(t). Then we seek the values
of o:1, a2, et3 that minimize
I: [A. (t) - o:,t - a2t - et3t ] .
k
SS ==
0
2
3 2
(12.16)
1=1
i�
The least-squares estimates are the values of o:1, a2, a3 that satisfy the sim­
ultaneous equations
= 0 for i = 1,2,J. The following example illus­
trates the procedure.
Parametric Alternatives to Exponential Models
307
j!EXAMPLE 12.4![ The following grouped data, measured in thousands of
miles, give the time to failure of I 04 bus motors. Preliminary examination
of the data suggests a quadratic hazard rate function. Find the least-squares
estimates of the parameters of this model.
TABLE 12.5
Time Interval Number of
of Failure
Failures
0-20
19
20-40
13
13
40-60
15
60-80
80- 100
IS
100 - 120
18
120+
11
!SOLUTION! The number of survivors, observed proportion surviving, and
observed cumulative hazard are given in Table 12.6.
Duration t
20
40
60
80
100
120
TABLE 12.6
Number Observed
S0 (t)
Surviving
85
.81730
72
.69230
.56730
59
.42307
44
29
.27884
.10576
11
Observed
A0 (t)
.20174
.36773
.56686
.86021
1.27712
2.24658
The familiar equation set to solve for lit, li2, ch is
0:1 I:;t 3 + o:2I:;t 4 + o:3:[;t 5 - E t 2 · A0 (t),
(l2.17a)
oq :[;t 4 + o:2:[;t s + a3:[;t 6 = I:; t 3 . Ao(t)
which can be written in the convenient matrix-vector form as
where
(t't)a = t'y,
(l2.17b)
Additional Miscellaneous Applications
308
f
y' = [A0 (20), A0 (40), · · ·, A0 (120)],
and
t=
20
40
(20)2
(40)2
(20)3
(40)3
1io (120)2 odo)3
J
The solution to Equation (12.17) is �1 = .0159148, th = -.0002545, and
c:l'.3 = .0000023.
0
12.3 NON-PARAMETRIC MODELS
Non-parametric estimation of a survival model is generally accomplished by
either the product-limit method or some version of the actuarial, exposure­
based, method. An illustration of each of these non-parametric approaches is
presented in the following two subsections.
12.3.1 Product-Limit Approach
The reader is expected to be familiar with the steps involved in estimating
S(t) by the product-limit, or Kaplan-Meier, approach from the description of
the method presented in Chapter 7. Here our objective is to illustrate how
the method might be applied in a clinical study, and to describe several
interesting extensions of the method.
Consider a sample of cancer patients who reached a state of remis­
sion through treatment by chemotherapy. Following the onset of remission,
which defines time t = 0 for each patient, one subsample continued to re­
ceive maintenance chemotherapy whereas the other ( control ) subsample did
not. The objective was to see if maintenance chemotherapy prolonged the
time until relapse (i.e., increased the length of remission). Thus in this study
the concept of survival is represented by continued remission, and the con­
cept of failure is represented by relapse to the cancer condition.
Consider the subsample of lO persons whose chemotherapy was
maintained. Suppose their lengths of remission (i.e., times until failure by
relapse) were 9, 13, 13+, 18, 23, 28+, 31, 34, 45+, 48. The times denoted
13+ , 28+ , 45+ represent the lengths of remission at which the persons were
Non-Parametric Models
309
lost to observation, so that the data are censored. Furthennore, note that
both a failure and a withdrawal (person lost to observation) occurred at the
same recorded time 13.
!!EXAMPLE 12.611 Use the Kaplan-Meier method to estimate the survival
function S(t) for the sample with maintained chemotherapy.
!SOLUTION! We first represent the data on the following diagram, where
the X's denote failures and the arrows denote withdrawals.
x I x
10
l
x I x
20
l
Ix x
30
40
l
x
!FIGURE 12.61
As stated in Section 7.6.3, in clinical studies it is customary practice to par­
tition the time axis at each failure point. Then the risk set immediately pre­
ceding thefh failure is calculated, leading to an estimate ofthe probability of
failure over the interval from the (j-1Y' to the fh failure. The estimation of
S(t) is summarized in the following table, where '1 denotes the time oftheJ'"
failure. Note the convention ofconsidering the death at time 13 to occur just
before the withdrawal at that time.
TABLE 12.7
j
1
2
4
5
7
8
10
lj
9
13
18
23
31
34
48
Risk Set
rj
10
9
7
6
4
3
1
A
qi
1/10
1/9
1/7
l/6
1/4
1/3
1
A
P;
9/10
8/9
6/7
5/6
3 /4
2/3
0
A
S(tj )
.900
.800
.686
.571
.429
.286
.000
(Note that times t3 = 13, t6 = 28, and !9 = 45 are withdrawal times rather
than failure times.) Recall that for all t such that tJ ::::; t < tJ+ l, the estimate
A
of S(t) is the same as S('J), since there are no deaths in the sample between tJ
and t. The estimated survival function is graphed in Figure 12.7.
0
Additional Miscellaneous Applications
310
S(t)
k
1.000 -------.
L
�
.750
.500
L
.429
.286
.250
0
10
20
30
40
!FIGURE 12. 71
Having estimated S(t) by the product-limit method, many clinical
studies will summarize the survival pattern by estimating the mean survival
time. (In our example, the mean survival time represents the average length
of the remission period before relapse.) The estimated mean survival time is
calculated as the area under the estimated survival function curve, and is
given by
µ = L S(tj_ i) · (tj-lj-1 ),
j=I
(12.l&)
Non-Parametric Models
311
where t1 is the time of the fh failure and m is the total number of failure
times. Note the convention that t0 = 0 and
S(to) = S(O) = 1.000.
(12.19)
!!EXAMPLE 12.7!f Estimate the average length of remission for the sample
with maintained chemotherapy.
!SOLUTION! Using Equation (12.18), or by simply referring to Figure 12.7,
we have
µ = 1.000(9) + .900(4) + .800(5) + .686(5)
+ 571(8) + .429(3) + .286(14) = 29.889.
Note that Equation (12.18) presumes that the last event is a failure rather
than a withdrawal. If this is not the case, then the estimate is bias.ed on the
0
�WS�.
µ
The variance ofµ can be estimated by
.,.__
n-1
A?1
t:f (n-j)(n-j+l)'
Var(µ") - �
(12.20)
where the summation is taken over the values ofj corresponding to failure
times, and A1 is the area under the curve of S(t) to the right of t1. Since
On the
to = 0, then A o is the entire area under the curve, so that A o =
I\
other hand, In is the time of the last death, at which time S(tn ) = 0, so that
clearly A n = 0.
µ.
IIEXAMPLE 12.S!f Estimate the variance of the average length of remission
for the sample with maintained chemotherapy.
!SOLUTION! The values of A1 are easily calculated from right back to left.
Referring to Figure 12. 7 we obtain the values
As = .286(48-34) = 4.004,
A 1 = Ag+.429(34-31) = 5.291,
As = A 1 + .571(31-23) = 9.859,
A4 = A s +.686(23-18) = 13.289,
A2 = A4 + .800(18-13) = 17.289,
Additional Miscellaneous Applications
312
and
A1 = A2+ .900(13-9) = 20.889.
(As a check we note that Ao= µ = A1 + l.000(9-0) = 29.889, which
agrees with the result of Example 12. 7.) Then using Equation (12.20) we
have
__......
/\
Var(µ) =
2
2
2
(20.889) (17.289) (13.289)
(9)(10) + (8)(9) + (6)(7)
(5.291)2
(9.859)2
(4.004)2
= 21.449.
+ (5)(6) + (3)(4) + (2)(3)
0
An alternative approach to calculating the product-limit estimates in
the presence of withdrawals has been suggested by Efron [24]. In a sample
of size n, if all departures from the sample were failures (i.e., if there were
no censored observations) then the graph of S(t) would step down at each
failure point. Efron refers to this quantity as the failure mass at that point.
Now if a particular departure is a withdrawal (loss from observation) rather
than a failure, then we know that the failure time for that sample member is
somewhere to the right of the observed withdrawal time. Efron proposes a
redistribute-to-the-right procedure, whereby the failure mass associated with
a particular withdrawal time is redistributed equally over all future departure
times (failures and withdrawals alike). When the next withdrawal time is en­
countered, the total failure mass distributed at that point is then redistributed
equally over all future departure times. This procedure is continued until all
failure mass has been redistributed and brought into the estimation of S(t).
We illustrate Efron's procedure with our sample of 10 persons with
maintained chemotherapy. If all 10 departure times were failures, theh each
/\
failure mass would be .10, and the successive values of S(t) would be 1.000,
.900, .800, and so on. In our sample, however, the yd departure time,
t3 = 13, is a withdrawal. Its failure mass of .10 is redistributed over the re­
maining 7 departure times, assigning �(.10) � .01428 to each future time, so
that each future time now has its original failure mass of .10 plus the redistri/\
buted amount, for a total of .11428. Now the graph of S(t) will step down
.11428 at each of t4 = 18 and t5 = 23. When we reach the next withdrawal
time at t6 = 28, the total mass of .11428 is redistributed over the remaining 4
departure times, so that each one now has .11428 !(.11428) = .14285.
i
/\
+
Then S(t) steps down by .14285 at failure times f? = 31 and ts = 34. Finally,
Non-Parametric Models
313
the total mass of .14285 assigned to the last withdrawal time t9 = 45 is redis­
tributed over the one remaining failure time at t = 48, for a total of .28570 at
that point. As a check on the arithmetic of this procedure, we have succes­
sive step-downs at our 10 departure times (withdrawal as well as failure) of
.10, .10, 0, .11428, .11428, 0, .14285, .14285, 0, .28570. These 10 values
sum to .99996, which differs from 1.00 only due to the rounding involved in
the division by 7.
Efron's interesting procedure is explored further in the exercises.
12.3.2. Actuarial Approach
There are many examples in the clinical and reliability literature of non­
parametric survival model estimation in which failure rates are first esti­
mated by the generic approach of "failures divided by exposure." In the
literature, this actuarial approach is also referred to as the life-table method.
As one particular illustration of the influence that actuarial science has had
in related fields, we choose the following example of constructing a life
table for light bulbs given by Benjamin and Pollard (11].
A large municipality wants to have a model from which to project
the expected cost of replacing light bulbs in its street lighting system over
future annual budgeting periods. Its current lighting system has been in place
for 6 years, and full records of bulb replacements over that time exist.
The first step in constructing the desired model is the calculation of
historic bulb-failure rates. An appropriate unit must be selected for the
length of the age interval over which such rates will be determined. For thi�
project an examination of the data has suggested that a good unit to use will
be a quarter of a year. (As an aside, it should be noted that hours of use is
likely to be a more appropriate measure of bulb age than is quarters since in­
stallation, but it would be very difficult in practice to have such information
for individual bulbs.)
We assume that the following information is available:
(I) The total number of street lamps in the system (denoted P).
(2) The total number of bulbs installed during the six-year history of
the system (denoted N).
(3) The number of failures that occurred after x but before x+ 1
quarters of life (denoted Bx ).
(4) The number of bulbs still in use at the end of the 24-quarter ob­
servation period that were installed between x and x+ I quarters
ago (denoted Px ). In the terminology of Chapter 6, these bulbs
constitute enders to the study.
Additional Miscellaneous Applications
314
Note that P represents the total number of bulbs in place at any point in time,
whereas N is the total number ever installed. N exceeds P because of fail­
ures and replacements. In fact, since
e
(12.21)
denotes the total number of failures, and hence the total number of replace­
ments, then it follows that
N = P+8,
(12.22)
since the overall total number of installations is made up of the total initial
installations (P) plus the total replacements (8). Furthermore, the total num­
ber in use at the end of the study should equal the number of lamps in the
system, so that
23
P - LPx.
x=O
(12.23)
The relations given by (12.22) and (12.23) can be used as a check on the
tabulations of the data.
In order to determine exposure for each quarter of age (i.e., each
estimation interval), we will assume that bulbs still in use at curtate age x are
age x+!, on the average. Then the "bulb-quarters" of exposure in the age
interval (0, 1) is given by
Eo = N- !Po,
(12.24)
the deduction reflecting the idea that the Po bulbs still in use at the end of the
study were exposed to the risk of failure for only half of the estimation
interval (0, 1).
For the estimation interval (1, 2), the total installations N are not all
exposed to the risk of failure in (1, 2). The Bo that fail in (0, l) and the Po
that are not yet age l at the close of the study contribute no exposure at all to
(1, 2). Furthermore, as established above, the P1 still in use are exposed for
only half of the (l, 2) interval. Thus the total exposure for (l, 2) is given by
Non-Parametric Models
315
- Eo - Bo - !(Po+P1 ),
(12.25)
since Eo = N-!Po by Equation (12.24). This shows us that exposure in
successive estimation intervals can be calculated recursively by the relation­
ship
(12.26)
for x = 0, I, ... , 2 2.
Once the exposure values are determined, the failure rates can then
be estimated as
(12.27)
for x = 0, I, ... , 23. As explained earlier in the text, it might be necessary to
graduate these initial estimates to reduce the effects of sampling error.
Finally a suitable radix is chosen and life table values of fx and dx
are determined by the familiar relationships. The construction of a tabular
survival model for this type of light bulb is now completed.
12.4 E')(]!.RC:ISES
12.1 Introduction
12.2 Parametric Alternatives to Exponentia! Models
12- 1. Let T have the piecewise-exponential distribution defined by Equa­
tion (12.l ) . If m denotes the median of T, find the relationship among
y, ,\ 1, and ,\2 such that m > y.
12- 2. Find the mean of the piecewise-exponential distribution defined by
Equation (12.1).
Additional Miscellaneous Applications
316
12-3. Find /(t), the PDF of the composite hazard rate model defined by
Equation (12.5), and show that
x
i f(t) dt = l.
12-4. Find the median of the Rayleigh distribution defined by Equation
(12.9).
12-5. Derive Equation (12.13).
12-6. Derive Equation (12.14).
12-7. Rework Example 12.3 assuming the observations are censored after
the first 15 failures.
12-8. Verify Equations (l2. l 7a) and (l 2.l 7b).
12-9. Calculate the chi-square test statistic for the estimated survival model
of Example 12.4.
12.3 Non-Parametric Models
12-10. Show that ifthere are no censored observations, then Equation (12.18)
reduces to the sample mean
µ = t =
I\
'5:, lj
j=I
n
12-11. Show that if there are no censored observations, then Equation (12.20)
reduces to
�
"
Var(µ) -
12-12. Use Greenwood's formula to estimate the variance ofthe estimate of
S(25) for the sample with maintained chemotherapy.
Exercises
317
12-13. Using Efren's redistribute-to-the-right procedure, calculate the values
of S(t) for the sample with maintained chemotherapy, and compare
them with the values shown in Table 12. 7.
APPENDIX A
PROPERTIES OF ESTIMATORS
In this appendix we give formal definitions of the various properties of
estimators which we have cited throughout the text. The extent to which
specific estimators do or do not satisfy these properties has been noted in the
text when such specific estimators were defined, and reference has been
made to other sources for further information regarding the properties of
specific estimators. In particular, Broffitt [14] gives a good analysis of the
properties of the actuarial estimator, the product limit estimator, and the full
data MLE under the exponential assumption.
It should always be kept in mind that estimators are random vari/\
ables. In this appendix we use(} to denote the estimator random variable for
the quantity(} which we wish to estimate. Note that (} could be a parameter
of a parametric survival model (e.g., ). of an exponential model), or it could
represent a qx value of a tabular survival model.
A.1 BIAS
/\
Since(}, the estimator of(}, is a random variable, some useful information is
/\
supplied by its expected value E [9]. Of particular interest is the question of
/\
how E[B] compares of(}. If
/\
E[B]
/\
= B,
(A.I)
then B is said to be an unbiased estimator of B. Alternatively, if
/\
E[B] > B,
(A.2)
/\
then(} is said to be positively biased, and if
/\
E[O]
< B,
(A.3)
/\
/\
then(} is said to be negatively biased. In general, the bias of(} is given by
/\
Bias(B)
= E [B] - 0.
/\
(A.4)
Properties of Estimators
320
/\
In some cases E[O] is a function of the sample size, n, from which the esti"
mate of O was produced. In these cases we do not strictly have E [O] = 0, so
I\
(} is not strictly unbiased. However, if it is true that
/\
limE[O]
n-+oc
= 0,
(A.5)
I\
then we say that O is asymptotically unbiased.
A.2 MEAN SQUARE ERROR AND VARIANCE
Another characteristic of an estimator that gives some information about its
quality is the extent to which the outcome of an estimator differs from the
I\
true value e which it seeks to estimate. Remember that the estimator, 0, is a
random variable, whereas the true value of O is fixed, albeit unknown. We
I\
define the expected squared difference between O and O to be the mean
I\
square error of the estimator That is,
e.
/\
MSE(O)
= E [(0- 0) ],
/\
(A.6)
2
where the expectation is taken with
respect to the random variable e. We see
.
I\
/\
that MSE(O)
gives an average, or expected, squared departure of O from e, a
priori to a consideration of actual data.
I\
A similar a priori measure of departure of() from () is given by the
variance, defined as
/\
Var(O)
/\
= E[(O -E[0])2 ].
/\
/\
(A.7)
There is a convenient relationship between the rnean square error
I\
I\
and the variance of 0. Beginning with ( A.6), we subtract and add E[O] inside
the parentheses to obtain
MSE(B)
= E[(B-E[O]+ E[0]-0)2 ]
/\
= E (0-E(0])2 + 2(E[O]--B)(O-E[O])
+ (E[0]-0)2
[/\
/\
/\
I\
/\
]
2]
/\
I\ 2 ]
I\
/\ l
r
I\
= E [(O-E[8])
+2(E[O]-O
)·E ["O.:.·E[O]
J+El(E[0]-0) ,
321
Properties of Estimators
I\
since 2(£ [O] - 0) is a constant. In the middle term,
since E [E[O]]
= E[O]. Thus we have
= E[(O-E[0])2 ] + E[(E[0]-0)2 ]
MSE(O)
= E[(O-E[0])2] + (E[0]-0) ,
2
(A.8)
since E [O] - 0 is a constant. Finally, by using (A.7) and (A.4), we see that
(A.8) becomes
/\
MSE(B)
= Var(O) + [Bias(O)r.
(A.9)
A.3 CONSISTENCY
Instead of looking at the expected difference between an estimator and the
quantity which it estimates, let us consider the probability that such differ­
ence (in absolute value) is less than a given number €. That is, we wish to
/\
consider Pr{IO - 01 < E}.
/\
Now let On represent the estimator of O when the sample from which
I\
it is derived is of size n. As n increases, it would be desirable for On to
become closer to 0. To investigate this, we will consider the probability that
I\
I\
On is close to 0, which is the same as the probability that 1011 - 01 < E. If this
/\
.
probability approaches one as n approaches infinity, we say that On 1s a
/\
consistent estimator of 0. Formally, On is a consistent estimator of() if
lim Pr{[Bn - Bl < E}
n-i>X
for any E.
= I
(A. IO)
322
Properties ofEstimators
A.4 EFFICIENCY
An efficient estimator is one which has a variance as small as, or smaller
than, the variance of any other estimator in the class with which it is being
compared. The class could be all consistent estimators, all asymptotically
unbiased estimators, and so on. In other words, an efficient estimator is one
that is of minimum variance within the class.
A.5 DISTRIBUTION
/\
In some cases the actual distribution of the estimator random variable () can
be determined. A simple example of this is the binomial proportion estimator. This discrete random variable takes on the values Q,
l, · · ·, �n, with
n n
probabilities given by a binomial distribution with parameters n and 0.
/\
Generally the precise distribution of () for finite n is extremely
complex. In many cases, however, this distribution approaches a normal dis­
"
tribution as n -t oo. If this is so, we say that () is asymptotically nonnally
/\
distributed, or that (} possesses the property of asymptotic normality. Broffitt [14] shows that the actuarial estimator, the product limit estimator, and
the full data, exponential distribution MLE are all asymptotically normal.
In discussing asymptotic distributions, it is customary to consider the
random variable
(A. l l)
/\
[f (} is asymptotically unbiased, its asymptotic mean is (}, so the asymptotic
mean of Zn is 0. Then if Z,, is asymptotically normal, its asymptotic
distribution is defined by its asymptotic variance.
Derivation of asymptotic variance can be quite complex, and we will
consider it only for maximum likelihood estimators. This is pursued in Ap­
pendix B.
APPENDKXB
ASYMPTOTIC PROPERTIES OF
MAXIMUM LIKELIHOOD ESTIMATORS
In this appendix we discuss the general conditions under which maximum
likelihood estimators will possess the asymptotic properties of consistency,
unbiasedness, and normality. We will state these conditions and results with­
out proof. The interested reader is referred to Rao [61] or to Rohatgi [64] for
further details. The development below follows that given on pages 384-385
ofRohatgi.
We assume an exact age at death model so that the likelihood for
one observation is the PDF, f(x;O). We use this notation to remind us that
the PDF is a function of both(}, the parameter whose MLE we seek, and x.
B.J CONDITIONS
Consider the following five conditions to be satisfied by f(x;(}):
(i) ln/(x;O) is thrice differentiable with respect to e. That is, the deriva.
.
8
8 2 lnf(x;O), and 83
lnf(x;O),
t1ves
883 ln/(x;O) all exist.
88
882
(ii)
J�00 %8 f(x;B)dx = 0
{This is the same as E[f8 ln/(x;B)] = 0.)
(iii) f�x :;2 f(x;O) dx = 0
x 8
lnf(x;O) . f(x;O) dx < 0
(iv) - 00 < L x ae2
(Note that the integral represents E ( 2 lnf(x;O)]. The purpose of
the condition is to establish that this expectation exists.)
2
(v)
There exists a function H(x) such that
E[H(x)] < oo.
g;
I g;3
I
lnf(x;O) < H(x), where
324
Asymptotic Properties of Maximum Likelihood Estimators
Let 8 be an open interval, on the real number axis, which contains the true
value of
Conditions (i), (ii), (iii) and (v) need hold only for all in 8,
whereas Condition (iv) is to hold for all 0.
e.
e
B.2 RESULTS
lf f(x;O) satisfies the five conditions, then we have the following results for a
a sample of size n:
(a) The likelihood equation has (with probability one) at least one solution
I\
(i.e., 0 exists).
/\
/\
(b) There exists a sequence of solutions, {On }, with On denoting the member
of the sequence for sample size n, that is a consistent estimator of e.
/\
That is, Pr{IOn - 01 <£}approaches one as n approaches infinity.
(c) As n-+ oo, the distribution of On approaches a normal distribution (i.e.,
On is asymptotically normal), with
/\
I\
(i) E [On ] = 0 (i.e., On is asymptotically unbiased), and
(ii)
Var(Bn ) = { n · E[
(J8 tnf(x;0)) ] }-l
2
Note that an equivalent way to express (c) is to say that Zn =
asymptotically normal with mean O and variance {
Jn (on -0) is
E[ ( to lnf(x;O)r] }-I
It should be kept in mind that the On referred to above is a solution of
the likelihood equation, but not necessarily an MLE of e, although in most
cases it will also be an MLE.
B.3 DERIVATION OF EQUATION (7.61)
We now need to show that the expression for Var(On ) given by (c), (ii) is the
same as that given by Equation (7.61) on page 165. That is, we wish to show
that
[ fl
.]
E -{)()2 In L(x; O)
n·
E[ (Se ln/(x;O)) 2].
(8.1)
325
Asymptotic Properties o/Maximum Likelihood Estimators
We note first that the likelihood is given by L(x;(}) = fr f(x;;(}), so
that In L(x;(}) =
we have
t
i=l
i=I
lnf(x;;(}). Since both E and 0�2 are distributive operators,
j(x;;(J) is the same PDF for all i, so E[-
J:1 lnf(x;;B)] is the same for all i.
Thus the left side of (B.1) is n · E [- ;;2 lnf(x;(J)], so it remains for us to
show that
Beginning with the left side of(B.3), we see that
g;2
Inf(x;(}) ==
g(} [ g(} Inf
(x;O)]
_ _Q_ [J'(x;(})]
8() f(x;B)
=
f(x;(}) · J"(x;(}) - [f'(x;(})] 2
2
[f(x;(})]
where f '(x;B) = J8 f(x;B) and f "(x;(}) =
[
]
2
E - a 2 lnf(x;B)
{)(}
(B.4)
J;2 f(x;(}). Then
2
[f'(x;(J)] - J"(x;(})
= E [ (x;())J
f(x;(}) ]
2
[f
= E
2
f"(x;O)
ae
lnf(x,
O) ) ] - E [
.
a
[(
f(x; O) ] ·
(B.S)
326
Asymptotic Properties ofMaximum Likelihood Estimators
Finally we see that the first expectation in (B.5) is the right side of (B.3), and
the second expectation is equivalent to f�x
Condition (iii) holds.
g;2 f(x;O) dx, which is zero if
APPENDJIXC
DERIVATION OF EQUATION (8.49)
In this appendix we show how the conceptual definition of the A2 statistic,
given by Equation (8.47) is evolved into Equation (8.49), the formula from
which A2 would actually be calculated.
We begin by noting the definition of S°(t), the observed distribution
defined by (4.1). Substituting this expression into (8.47), we have
A
2
=
t
-t
j'i+I
S(t) �F(t}
[ n ;;- i - S(t)rf(t)dt
00
+ f.
S(t) 2f(t) dt
S(t) �F(t) [O- ]
(t)
(t)] ]f(t)dt
S(t) �F(t) [(n; ) -2(n; ) ·S +[S
i
j'i+l
= 1
00
2
(C.l)
i
oo
2
[S(t)] j(t)dt
+ /. S(t) �F(t)
t
2
S(t) �F(t) [S(t)]
+
f(t)dt
2
i
; 1
)
j' + S(t) �F(t) [ ( n;
2
- 2( n; ) . S(t)] f(t)dt,
i
(C.2)
by combining the two terms involving [S(t)] 2 into one integral. Then (C.2)
simplifies to
A2
= n. {
Jo
x,
.
S(t) /(t) d
t
F(t)
n-1
11 + /(()
2(n i) · t; ; 1 F(t) dt
� -
n-1 n-i 1 1';+1
j(t)
)
(
dt
n ·
+
F(t) .
(t)
S
L
i==O
I;
·
(C.3)
328
Derivation of Equation (8.49)
Consider the three integrals in (C.3). Let the indefinite integrals be
represented by
S(t) · f(t)
dt
Ii =
F(t)
J
--
Ii
f
Mdt
F(t)
Integration by parts produces /1 = lnF(t) - F(t). Straightforward integra­
tion produces Ii = ln F(t). Integration by partial fractions is used to produce
h = In F(t)-lnS(t). Substitution of these results into (C.3) produces
A2
= n[ln F(t) - F(t)],:
n-1
- L 2(n-i)[lnF(t)] 1';+I;
i=O
n-1 0.=Jf
z
1
L n-;, [lnF(t) - lnS(t)JIt
+
i=O
i+t .
I;
(C.4)
Next we simplify the three terms in (C.4). For convenience we call
them A, Band C, so thatA2 = A-B+C.
A:
This easily simplifies to n[lnF(oo)- F(oo)- lnF(O}+F(O)]. Recall
thatF(oo) = l, F(O) = 0, and ln(l) = 0. Thus we have
A
= - n - n · lnF(O).
(C.5)
We notice that F(O) = 0 so we have the term ln(O), which is awkward
to define. For now we will retain the notation lnF(O).
n-1
B: This term simplifies to 2 · I: (n-i) · [lnF(t;+1) - lnF(t;)], which is
i=O
2 [n · lnF(t 1 ) - n · lnF(to) + (n-1) · ln.F(t2)
-(n-1) · lnF(t1) + ··· + lnF(tn)- lnF(tn-1)],
which simplifies to 2[lnF(t1) + lnF(t2) + ··· + lnF(t,, )- n · lnF(t0)].
Derivation ofEquation (8.49)
329
Thus we have
= 2 · L lnF(t;) - 2n · lnF(O),
n
B
i=l
(C.6)
since to = 0.
C: This term can be written as
� · Z:: (n-i) • [lnF(t;+1)- lnF(t;) + lnS(t;)- lnS(t;+1)].
n-l
2
i=O
Expanding the summation we have
�{n2[lnF(t1)- lnF(O)+ lnS(O)- lnS(t1)]
+ (n-1)2 [lnF(t2) - lnF(t1) + lnS(t1) - lnS(t2)]
+ · · · + 1 2 [lnF(tn)- lnF(tn_i) + lnSCtn-1)- lnS(tn)]
+ 02[lnS(tn)- lnF(tn)]},
(C.7)
where the zero-value last line is added for symmetry. Now each of the
terms lnF(t;) and lnS(t;), t = I,- ··,n, appears twice in (C.7). Grouping
these terms we find that (C.7)becomes
k(lnF(t 1){n2 - (n-1)2 } + lnF(t2){(n-l)2 - (n-2)2 }
+ ··· + lnF(tn){l2 -02 }]
- MtnS(t 1){n2 -(n-1)2 } + lnS(t2){(n-1)2 -(n-2)2 }
+ · · · + lnS(tn){1 2 -02 }]
+ n · lnS(O)- n · lnF(O).
Since S(O) = 1 and ln(l) = 0, this expression can be written as
330
Derivation of Equation (8.49)
h·L
[(n-i+ 1) - (n-i) ] · lnF(t;)
i=l
- (i-1) ) · lnS(tn-i+t)- n · lnF(O).
- h · :Z::)P
i=I
fl
2
2
n
2
(C.8)
Next we make the simplifications [(n-i+1)2 - (n-1)2 ] = 2(n-i)+ 1
and (i2 - (i-1}2 ] = 2i - 1. Finally we can write term C as
C
= - n · lnF(O) + k · L (2n-2i+1) · lnF(t;)
n
i==l
(2i-1) · lnS(tn-i+t).
- k·L
i=I
n
(C.9)
Combining A 2 = A - B+C, we have
A2
n
= - n - n · lnF(O)- 2 · L lnF(t;) + 2n · lnF(O)- n - lnF(O)
i=I
+ k · L(2n-2i+I) -lnF(t;)- k · L(2i-1)· lnS(tn-i+I)
n
n
bl
bl
= - n- h · L
n
i=I
=
- n-
(2i- l) · lnF(t;)-
h · L (2i-1)· lnS(tn-i+l)
n
i=l
(2i-l){ln[F(t;) · S(t -i+I )}
] ,
k·L
i=I
n
n
(C.10)
/\
which becomes (8.49) when F(t;) and SCtn-i+I) are evaluated by F(t;)
/\
and S(tn:i+I ), respectively.
APPENDIXD
EQUIVALENCE OF EQUATIONS (I0.22a) AND (I0.22b)
In Panjer's model, µi denotes an estimate of the constant force of progres­
sion from Stage j to Stage j+1 of the disease. The available data show only
the number of patients who were in Stage j of the disease when first
observed, and the number of them who progressed out of Stagej while under
observation. Since members of the sample were observed for different
lengths of time, with no record of the precise time of observation for each,
we assume that all were observed for the (fixed) average of the reported
times. Thus our model is binomial for the number progressing out of Stage j
within a fixed time t, with a constant force of progression assumed to apply
while in Stagej. We suppress the subscript/ for convenience.
Under the constant force assumption, the probability of remaining in
Stage j to the end of observation is p = e-µ.,, and the probability of progres­
sing out of Stage j is q = 1-e-µ1• If n persons are observed, and d of them
progress out of Stagej, then
(D.1)
and the log-likelihood is
f = lnL(µ) = d·ln(l-e-µ.')+(n-d)(-µt ).
(D.2)
Then the likelihood equation is
of = d·t·e-µt1 - t(n- d) = 0,
8µ
l - e-µ.
from which we find
d · t ·e-µt = t (n-d)(l-e-µ.1),
so
d·e-µt = n(l-e-µ1) - d + d · e-µ,,
(D.3)
Equivalence ofEquations (J0.22a) and (W.22b)
332
and finally
= n-d
e -µt = I - d_
n
n .
Next we find
= (l-e-µ1)(-dr. e-µt) -1 (2dt. e-µt)(t. e-µt)
[}2£
aµ,2
(D.4)
=
=
(l-e-µ )
2
1
2
t2
.
t
e
µ
d
d . e-µ1 - dt2 . e-2µ1
-dt2 · e-µt
(l-e-µ.1)2 ,
(1-e-µr)2
so that
(D.5)
The random variable for the number progressing out of Stage j, D, is binomi­
al, so E[D] = n(l-e-µ1). Then
(D.6)
We wish to show that Equations (D.5) and (D.6) produce the same
result when evaluated atµ=µ. From (D.4) we see that fl is the value ofµ
such that e-µt = n;d , where n and d are sample data values. Then, from
(D.6),
[ a e] !
E -2
=
2
{)µ
/F/J-
2
n?(";,d )
= ?(n-d) = nt (n--d) .
I - n-t!..
d
d/n
ll
(D.7)
On the othei hand, from (D.5),
-
a2 e
8µ2 I µ=µ
=
dr( n;,d)
=
2
( I - " d)
;,
! . t (n-d) = nt (n-d) .
2
(dj,i)-2 -
2
d
(D.8)
The equivalence of (D.7) and (D.8) establishes the equivalence of (I0.22a)
and (10.22b).
ANSWERS TO THE EXERCISES
CHAPTER2
2-1
(a) .00625
2-2
2
(a) e-(a1+b1 /2)
2-3
45V2
2-4
Because
2-5
I
3
2-7
In
-2
2-8
(a) constant
2-9
(a) (i) r
2-11
4
I
(b) .01
(c) .69315
(d) 66.66
2
(b) (a+bt)e-(a1 +b1 /2)
(e) 888.88
(c) b-1/2 - ab-I
lim S(t) = e-l/r =/= 0
/-+OC
.3
(b) decreasing
(ii) 2r
(c) increasing
(iii) r�2 , r > 2
(b) 4
I
2-12 4.7504
2-13
15.4822
2-14
x); truncation from above increases the hazard rate, so it
1 -e�-X(zmust become a function ofremainingpossible lifetime (which is z-x).
Note that the truncated HRF increases as x increases, i.e., as z-x
decreases.
2-15 (b) They are the same
2-16 Sy(y) = exp [ - n!1 · eY<n+I)]; Ay(y) = k · e><n+I) . L et k =Band
(n+l) = Inc, so that c = en+I . Then we recognize the Type 1
Least Value distribution defined in Exercise 2-15.
2-18 Fy(y) =
� 1;2, 0 '.Sy '.S oo
1r(l+ ) y
334
Answers to tlte Exercises
1:-
2-19 (a) 2l y1r; 1 - 4l 1r
(b) l. _41
. 8,
2-20 2; 1
2-21 (a) u2 . e2µ
2-22
(b) 3t
Exact: E[Y] = .4; Var(Y) = .04
Approximate: E[Y] = .4; Var(Y) = .0377
CHAPTER3
3-1
(a) S(x): l,.9,.72,.432, .1296,0 (b)
(c) w = 5, the first age for
which S(x) = 0.
3-2
(a) 5680
(c) .144
(b) .52
3-4
x
x+I - &
(a) S(x) - S(x+l) = e f;
- lo
(a) 90 (b) .2 (c) 5/24
3-5
2081.61
3-6
(a) - sPxµx+s
3-7
.0012749
3-8
(a) /5 (64-.8xt213
3-3
3-9
(a) µx Tx __ l
lx
(b) 60
(b) -}x'
3-12 2/15
{b) f,+n
= - Jex
.f.x
10,000
9,000
7,200
4,320
1,296
0
(d) 1.00
(b) sPx(µx -µx+s)
3-10 7.5; 8.03571
3-13 (a) - ndx
x
0
1
2
3
4
5
(c) 514.286
dx
1000
1800
2880
3024
1296
Answers to the Exercises
3-14 (a) Tx (b) Tx - Tx+n
335
(d) f\+n
(e) n · fx+n
3-18 (b) and (c)
3-19 .5214
3-20 2
3-22 Exact: .007843 l; Approximate: .0078905, .0079061, .0079217
3-23 13,440
3-24 .46
3-25 2.78084
3-26 .1(1-k)
3-27 All are correct
3-29 (a) 25.1
(b) 25.096819
3-3 I 150,000
CHAPTER4
4-1
7
4-2
4-3
3.75
71- 41
4-4
E[1];
4-5
I 1 / 16
4-6
- (,Jqo)(,lqo)
4-7
2'
2
Var(])
n
n
-I · S(r) · [1-S(t)]
n
Answers to the Exercises
336
4-8
9/10
4-9 .00009
4-11 . 000084
4-12 Reading down d, = 200, 150, 150, 50, 50, I 00, 100, 100, l00
each column: n1 = 1000, 800, 650, 500, 450, 400, 300, 200, 100
q, = .200,.188,.231,.100,.111,.250,.333,.500, 1.00
/\
S(t) = 1.000, .800, .650, .500, .450, .400, .300, .200, .100
-p, )
4- 14 (a) 2<tl+p,
(d)Biased
4-16 Exponential: In;= .040821;
Uniform:
In; = .040816;
Var(fn;)= .00041667
Var(fn;) = .00041631
4-18
CHAPTERS
5-1
Date ofjoining the pension plan.
5-2
(a)25.75
(b) 28
5-3
(a)31.75
(b) 31.75
5-4
x � y; < x+ l
5-5
X < Z;
5-6
(a) (.75, 1)
5-7
(a) Special Case B (b)Special Case A
5-8
.28
5-9
20 -- t
600
5-10 .18
�
(c) Undefined
x+ 1
(b) (0, 1)
(c) (0, .75)
(c)Special Case C
337
Answers to the Exercises
CHAPTER6
6-1
(s-r)·qx
I - r·qx
6-2
(s-r) ·qx
l -(l-s)·q,
6-5
6-6
.01487
6-8
(b) Positive
6-9
(a) gx(l-qx
ll
)
x
n
n
Ls; qx(l-s;qx)
L 1-r; qx+r;(l-1-r;qx+r;)
(C) '-i= I
(b) '-i= ----..,,-
--
I
[ti(l-r;f
-(l-qxY
qx
(b) l
6-10 (a) s
6-11 (b)
6-12
6-13
!
2
1=1
s
()
C 1 -(1-s)·qx
(b) l
1
qx + lnpx
d"
_=.i:..........
nx - Cx'
n
[ts;]
where d"
x
= dx - d'x
dx
1:(s;-r;)- f·wx
i=I
6-15
4
(a) b-};:- ad,wherea
and
b
(b)
b-
= [4n(l-r2)-4wr(l-r)+4dr(l+r)]
= [n(l-r) - !w(l'-r) + !d(l+3r)]
Jba 4aw , wherea = [ I n(l-r)
1 - I dr(l-r) + I w l +
2
2 r( r)]
2
Z
and b = [n(l-r) - td(l-r) + two +3r)]
2
-
Answers to the Exercises
338
(b)
b - Jb2 - 2ns4w where b = ns2 - -s
I 2
I 2
d+ -s
w
'
ns3
2
2
d,/(1-r)(d,+w,)
w,/(1-rXd,+w,)
d
d
6-18 (a) I - [nx- x-Wx ]
(b) 1 - [ n,- x-Wx ]
�
�
d,/s(d,+w,)
x x]
6-19 (a) 1 _ [nx-d-w
nx
6-20 Hyperbolic: .01739;
Linear: .01737;
[
]
,-Wx w,/s(d,+wx)
(b) 1 _ nx-d
nx
Exponential: .01738
6-21 .3439
6-22 m
A(W)
= 1/16;
A(W)
q
= 2/33
6-23 2/7
6-24 .25
6-26 (a) nx · qx
(b) (i) Cx • sPx · 1-sqx+s
6-27 (a) No. Dn and De are not independent (since De :'.S Dn ), so
Var(Dx ) = Var(Dn )+ Var(De )- 2 · Cov(Dn , De )CHAPTER7
7-1
d "'"' 2(1-si)
n
- 0
- I -q + q + � [ l - (1-s;)q ] -
7-2
d
n+d
7-3
fix = 0 if none die. fix = 1 only if all die and S; = 0 for all i;
thus, in general, iJx < 1 even if al! die.
Answers to the Exercises
7-4
339
(a) (1-qt-d · TI [(I-qt· - ln(l-q)]
'D
(b) (n-d) · ln(l-q) +LS;· ln(l-q) + L ln[ln(l-q)-1]
'D
'D
(c) I - exp [ (n-i:L s,]
v
7-5
2/151
7-7
(a) .60839
(b) .66667
7-8
(a) .05
(b) .05
7-9
(a) (i) a · qx (ii) I - a · qx (iii) 0
(ii) 0 (iii) I - qx
(c) (i) qx
(b)(a · qx )d; · (1-a · qx f,-d;
(d) (qx )d;' . (I-qx tx-c,-d:'
7-10
7-11 (a)
b - /b2 - 4rdx(nx-kx)
_ (n -rk +rd )
, where b x
x
x
2r(nx - k)
x
where d; is the number of observed deaths out of the kx entrants
at agex+r.
7-12
l + Px · lnpx
qx
7-13
I
7-15
+ _!k__
lnp,
Answers to tlte Exercises
340
7 - 18
-b - Jb�;J; , Wh ere b = 11x+ tdx - (1 -I) Wx
2tnx
7-19
k(l-r)
]-I
[ q(lb-q) + �
(I -rq)2 + q(l-q)(l-rq)2
µ2
7-20
7-21
7-22
q(l -q)( I - aq)
n(l-aq) - c( l-a)
7-23 (a)
(c)
d
nx · qi )
d
d+w
d+w
_ (d+w)k
( d+w)k.
w
d+w
- 132 + (o:+/3)2 - (l �k)2 '
(d+w)k.
(l-k)2
aTp(l-k)
(i)
a2
n(l-k)
nk
n(l-k)
nk
- (a.+fJ)2 + 1-k
ih<I -k)
(ii)
(e)
[In parts (c}, (d), (e), k = e-<a+/3l.]
- o:2 + (o:+.8)2 - (l-k)2 '
(a+/3)2
(d)
w
(b)nx · qi >
132
...
nk
n(l-k)2
( Ill) - (a.+fJ)
+I-k
- (a.+fJ)2 + I - k
From part (d),
- a.n(l-k) + ank(a+f3)
2(l -k)
Ji11 -_ n(l-k) (a+f3)o:(a.+f3)
2
fi[
2
2
_ £. _ - n(l-k)2 + nl'(o:+/3)2
2 - J2l (a+f3) 2(J-k)
_ n(l-k)2 (a.+f3) -f3n(l-k)2 + {Jnk(o:+(3)
/22 Let Ll
p(o:+/3)2(1-.k)
2
= Ji 1 · h2 - /12 · h I·
Then Var(8:) =
lg;
Var(/3) = !11 ;
I\
Ll
Cov(a, ,8) =
I\
I\
- /12
Ll
Answers to the Exercises
341
(b) .00042963
7-25 (a) .04227
7-26 2
7-27 3
7-28 (a) .20000;
1.60944
(b) .27711;
1.28333
7-29 .43926
CHAPTERS
8-1
,\2
8-2
Ca) ,; · Et;
8-3
.3
8-4
2/11
8-5
(a) 15.2
8-6
1.175
8-9
.68422
n
I
n
(c) .(:_
n
i=I
(b) 15.0
8-10 1.6
8-11 .07438
8-14 72.l
8-15 17.5
8-16
b
+ /3. E w.x(x+4)
b
+ /3 · E wx(x+!)
a
a:· E wx(x+f)
a
8-17
b
0: · L Wx
b
0: · L Wx
a
= E Wx . ln µ�+1/2
b
a
b
a
b
2
= E wx(x+!) · lnµ�+l/i
a
a
+ /3 · L Wx · ln(x+f) = LWx · In µ;+l/2
b
b
a
a
b
b
a
a
a· Ewx · ln(x+t)+/J· Ewx[ln(x+f)]
2
b
= LWx
· ln(x+!) · lnµ�+t/2
a
Answers to tlte Exercises
342
8-18 k = 2.1583 x 10-10;
,;i = 5.3884
/\
+de· I:wxBo(x+f )c�-:z
l
= Ewx (ux+f)
+de· EwxBo(x+})c�
= I:wxC�+f (ux+ 1)
dA · Ewx Bo(x+!)c�-1 + dB· EwxBo(x+})cf +de · Ewx [Bo(x+f )c�-ir
b
=
where E represents L, and ux+!2 -
= EwxBo(x+})c�-i (ux+!)
x=a
b
"2::x· w., · ln{I0,000µ:)
8-20 "��a __b____
"2::x2·wx
=
x=a
8-21
8.5;
approximately .039
8-22 28.45;
8-23
less than .0002
g
L [n · e-51\1-e-5.x)- sd s1] sd 51
x+
t=O
2
x+
8-24 x 2 = 13.8; reject with approximately 91% confidence
8-25 Reject with approximately 96.5% confidence
8-26 a =
- 45;
b = 37;
c =
- 10
8-27 .69563
8-28 1.58043
8-29 5556
8-31 .015740;
.001071;
8-32 53/54;
- 52/54
8-34 (a) Inc;
lnB;
1
.014322
(b) Itis
Answers to the Exercises
3_43
CHAPTER9
9-l
;
1
2
3
4
5
y,
Z;
();
¢;
29.72
33.50
31.56
39.72 -043.50 33.51
0
41.56
-00
32.48
28.09
38.09 30.07
0
****
**** ****
****
;
y;
Z;
();
¢;
36
37
38
39
40
31.34
31.38
30.52
32.16
31.22
37.50 -036.84 34.38
0
35.92
37.32
0
36.16
0
35.69
0
0
33.92
0
6
7
8
9
IO
32.80
31.35
29.60
30.85
0
42.80
0
41.35
0
39.52
40.85 39.21
****
0
3l.52
30.52
0
41
42
43
44
45
29.67
31.04
30.14
30.35
31.97
0
34.46
35.70 3l.75
0
34.43
0
34.08
35.99
0
0
0
32.94
0
0
11
12
13
14
15
32.04
31.62
29.82
31.28
30.ll
0
42.04
0
41.49
0
39.56
40.86 37.14
39.21
0
33.41
41.10
0
0
37.57
46
47
48
49
50
31.34
30.15
29.89
31.62
30.29
35.15
0
33.62
0
33.38
0
34.99
0
33.81 32.54
31.62
3l.l5
32.89
0
0
16
17
18
19
20
29.83
32.36
29.75
31.58
3l.86
38.72
0
41.39 39.17
0
38.23
0
40.18
0
39.90
0
0
34.49
0
34.55
51
52
53
54
30.37
30.41
32.26
29.69
31.76
33.27
0
0
33.13
35.32 33.74
0
32.30
0
34.11
32.24
0
0
30.02
0
21
22
23
24
25
30.00
31.39
29.73
31.62
29.53
38.05 35.23
0
39.06
0
37.42
0
38.93
36.45
0
0
0
0
32.62
36.33
56
57
58
59
60
30.08
30.59
30.66
32.22
31.84
0
32.37
32.98
0
32.74 31.37
0
34.41
0
33.91
0
0
0
32.31
0
26
27
28
29
30
30.04
31.97
31.65
29.90
30.60
37.25 32.12
0
39.32
36.70
38.78
0
36.73
0
37.60
0
36.74
0
0
33.07
61
62
63
64
65
29.69
31.20
29.94
32.20
29.70
0
31.68
32.84
0
31.38 31.06
0
33.49
0
30.61
30.10
0
0
33.37
0
31
32
33
34
35
30.80
29.56
31.32
30.08
30.47
0
37.40
0
36.ll
38.06 36.19
0
36.41
0
36.31
0
0
0
31.04
0
66
67
68
69
70
32.24
3L47
30.04
31.67
33.29
32.15
30.45
31.87
****
**** ****
55
****
0
0
0
0
**** ****
**** Member is not involved in the study.
33.24
0
0
0
****
Answers to the Exercises
344
9-2
9-3
9-4
x
30
31
32
33
34
35
36
37
39
Interval
(28,29]
(29,30]
(30,31]
(31,32]
(32,33]
(33,34]
(34,35]
(35,36]
(36,37]
(37,38]
(38,39]
(39,40)
(40,41)
(41,42]
(42,43]
x
30
31
32
33
34
35
36
37
39
dx
1
3
2
2
1
1
2
1
2
Exact
Exposure
.91
4.98
24.22
35.33
44.20
35.80
28.06
23.83
1627
10.53
8.72
4.72
2.18
1.10
.80
(a): qlj
.04045
.08141
.04424
.05433
.03501
.04110
.11567
.09060
.34540
Scheduled
Exposure
.91
4.98
25.15
36.53
45.54
36.55
28.68
24.60
17.38
11.39
8.72
6.34
2.18
1.10
.80
(b):�
.03922
.07595
.04444
.05479
.03448
.04167
.11111
.08333
.30770
Actuarial
Exposure
.91
4.98
25.15
37.15
45.54
36.55
28.68
24.60
17.38
11.39
8.72
6.34
2.18
1.10
.80
345
Answers to the Exercises
9-5
i
I
2
3
4
5
Zi
(Ji
</>i
0
32
28
37
29.95
0
34
*
43
0
0
30
32
39
41
0
39.62
11
12
13
14
15
33
32
30
31
30
42
41
39
40
39
16
17
18
19
20
30
32
30
32
32
21
22
23
24
25
Yi
Yi
Z;
(Ji
</>,
TI
31
31
32
31
37
36
36
37
35
-034.01
0
0
0
36
30
0
41
42
43
44
45
30
31
30
30
32
34
35
34
33
36
0
31.71
0
0
0
0
0
33
0
0
0
0
0
36.86
0
34
41
0
0
38
46
47
48
49
50
31
30
30
32
30
34
33
33
35
33
0
0
0
0
32.25
31
31
33
0
0
38
41
38
40
40
0
38.81
0
0
0
0
0
35
0
35
51
52
53
54
55
30
30
32
30
32
32
32
35
32
34
0
0
33.48
0
0
31
0
0
30
0
30
31
30
32
30
38
38
37
39
36
35.23
0
0
0
0
0
0
0
33
36
56
57
58
59
60
30
31
31
32
32
32
33
33
34
34
0
0
31.71
0
0
0
0
0
32
0
26
27
28
29
30
30
32
32
30
31
37
39
39
36
38
32.08
0
37.05
0
0
0
37
0
0
34
61
62
63
64
30
31
30
32
*
31
32
31
33
*
0
0
0
0
*
30
0
0
33
31
32
33
34
35
31
30
31
30
30
37
36
37
36
35
0
0
35.87
0
0
0
0
0
31
0
66
67
68
69
70
32
33
0
33
6
7
8
9
IO
*
32
*
*
39
*
41
*
*
*
0 -0*
*
i
36
37
38
39
40
30
*
*
*
*
*
*
65
*
*
*
*
*
*
*
*
* Policyholder is not involved in the study
*
*
*
*
0
0
34
0
*
*
*
*
*
346
Answers to tire Exercises
9-6
x
28
29
30
31
32
d_,
nx
33
34
35
36
37
1
1
2
1
35
29
23
15
11
38
39
40
41
42
I
1
0
0
0
7
4
2
1
1
ncxJ+t
dcxJ+t
n[x]+t
2
13
0
1
0
2
2
l
9-7
t
x= 30
d[x]+t
0
1
2
3
4
5
-00
2
0
1
22
18
16
11
IO
8
6
7
8
9
0
0
0
4
3
I
10
1
1
23
32
43
0
x = 31
0
0
1
I
ll
1
10
9
7
4
0
0
I
2
I
1
x = 32
d[x]+t
n[x]+t
0
1
6
5
l
4
3
2
I
-0- -1513
l
9
0
7
0
0
0
0
0
1
Answers to the Exercises
9-8
;
I
2
3
4
5
6
7
8
9
IO
Y;
Z;
347
B;
<p;
32
*
*
*
*
40
*
*
*
*
0
*
*
*
*
0
*
*
*
*
*
*
*
*
35
43
*
*
0
*
*
0
33
41
38.86
0
*
*
*
*
*
*
;
Yi
Z;
B;
-0-
<p;
36
37
38
39
40
31.34
31.04
30.10
32.34
31.56
38
37
36
38
37
41
42
43
44
45
29.71
30.84
30.21
30.77
31.48
35
36
35
35
36
0
31.55
46
47
48
49
50
31.69
30.03
30.01
31.13
29.98
36
34
34
35
34
31.97
0
0
31.03
0
33.01
0
0
32.23
0
35.69
34.04
0
0
0
0
34.10
0
0
0
0
0
0
0
33.01
0
0
11
12
13
14
15
*
*
34
32
33
32
42
40
41
40
0
36.78
0
16
17
18
19
20
31
34
31
33
32
39
42
39
41
40
0
0
0
39.28
0
34.76
0
0
0
34.15
51
52
53
54
55
30.60
30.78
32.44
29.89
32.15
34
34
36
33
35
0
0
33.92
0
0
21
22
23
24
25
39
40
38
30
31.19 39
29.58 37
35.68
0
0
0
0
0
0
0
32.19
36.38
56
57
58
59
60
30.21
30.1l
30.42
32.31
31.43
33
33
33
35
34
0
0
0
0
31.13
0
32.40
0
0
0
26
27
28
29
30
30.29
32.15
31.37
29.67
30.50
38
40
39
37
38
32.37
0
36.42
0
0
0
36.92
0
0
32.97
61
62
63
64
65
29.51
30.86
30.06
32.21
29.59
32
33
32
34
31
0
29.92
0
0
0
31.18
33.38
0
0
0
31
32
33
34
35
30.90
29.95
31.76
30.17
30.66
38
37
39
37
37
0
0
36.63
0
0
0
0
0
31.13
0
66
67
68
69
70
32.44
31.82
30.09
31.30
33.54
34
33
31
32
34
31
32
0
41.l l
0
0
37.86
* Member is not involved in the study.
0
0
0
0
0
32.47
0
0
30.22
0
33.45
0
30.53
0
0
348
9-9
Answers to tire Exercises
.09553
.04993
.02918
.03384
35
36
38
39
.04020
.14514
.10673
.14720
9-10
!
29
30
31
32
33
34
35
36
37
38
39
40
41
42
CHAPTER JO
10-1 4.20
10-3
10-4
qx
x
31
32
33
34
Fiscal Age
(a) Last Birthday
3.5
16.5
31.5
38.5
35.5
30.5
24.5
20
11.5
9
7
3
1.5
l
Calendar
(b) Fiscal Age
0
19
33
41
33
30
25
20
12
9
7
3
1
1
349
Answers to the Exercises
10-5 .15410
10-6 .92419
10-7
1- e-µ;
100 · e ILJ
10-8 (a) µac �µad
(c) l +
0
1
o {Pr[K1(t) = 0 I K1(0) = O] · (µac+µad)
J
r
CHAPTER JI
11-2
11-3
l
l+,·t6
6 -t
,[J. 1
(l +1··t 6 )2
11-5
/c5
([J�lr
11-6
.61880
11-7
.02927
11-8
.00001114
11-10 .00514
ll-1 l .00510
+Pr[K1(t) = II K1(0) = O] · (µbc+µbd)} dt
350
Answers to the Exercises
11-12
n
1
2
3
4
5
11-13
Cn
99,471.53
98,940.45
98,406.73
97,870.38
97,331.37
n
Cn
1
2
3
4
5
99,954.39
99,892.57
99,812.96
99,715.63
99,600.45
11-14 qt-I, Pt-I, and ,qo
11-15 (a) .04240
(b) .96381
(c) .31910
CHAPTER12
12-1
12-2
12-3 f(t) =
{ .X • exp[-.Xt]
[ .X + f,(t-y)2 ] · exp[ -.Xt- � (t-y)3 ]
12-4
- .Xo + J.X5 + 2.77259 .X1
-·
2.x,
12-7
>.o = .0597;
12-9
2.42
/\
>., = .0047
/\
12-12 .02678
12-13 They are the same.
o::;t<y
t >y
BIBLIOGRAPHY
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63. Robinson, J.M., Discussion of "Maximum Likelihood Alternatives to
Actuarial Estimators of Mortality Rates," by J.D. Broffitt, TSA, XXXVI
(1984), 125.
64. Rohatgi, V.K., An Introduction to Probability Theory and Mathemati­
cal Statistics. New York: John Wiley and Sons, I 976.
65. Ross, S.M., Stochastic Processes. New York: John Wiley & Sons, 1983.
66. Schwartz, D. and P. Lazar, "Taux de Mortalite par une Cause Donnee
de Deces en Tenant Compte des Autres Causes de Deces ou de
Disparition," Revue Inst. de Stat., 29:3 (1961),15.
67. Seal,I--1.L.,"The Estimation of Mortality and Other Decremental Proba­
bilities," Skand Aktuar., 1954, 137.
68. --,"Deaths Among Prospective Existings," PCAPP, XI (1962),308.
69. --, "An Attempt to Convert American Actuaries," The Actuary,
March 1981,4.
70. Shooman, M.L., Probabilistic Reliability: An Engineering Approach.
New York: McGraw-Hill, 1968.
71. Spiegelman, M., Introduction to Demography. Cambridge: Harvard
University Press,1968.
72. Stephens, M.A., "EDF Statistics for Goodness of Fit and Some
Comparisons," JASA, 69 (1974), 730.
73. --, "Goodness of Fit for the Extreme Value Distribution," Biome­
trika, 64 (1977), 583.
74. Tenebein, A. and I. Vanderhoof, "New Mathematical Laws of Select
and Ultimate Mortality," TSA, XXXII ( l 980),119.
75. Wolfenden, H.H., "On the Formulae for Calculating the "Exposed to
Risk" in Constructing Mortality and Other Tables from the Individual
Records of Insured Lives," TASA, XLIII ( 1942),234.
INDEX
Ackland, T.G., 132
Actuarial estimation,130-133
bias, 133
exposure,131-133
variance, 133
Additive models, 211-213
Age vectors, 227,232
AIDS, 248,249
AIDS-related complex (ARC), 249
Altman, E.I., 289, 291
Ananda,M.M.A., 299
Anderson-Darling statistic,
205-206, 327-330
Anderson, T.W., 205
Balducci,G., 65,133
Balducci hypothesis, 65, 138
Barlow, R.E., 300
Batten, R.W., xii, 65,132,133
Beekman, J.A., xii
Beers, H.S., 133
Benjamin, B., xii, 105, 313
Bias, 319-320
estimated survival function, 135
maximum likelihood estimators,
165,324
moment estimators, 119-12 J
produce limit estimator, 170
Bond default rates (see corporate bond
default rates)
Bond survival rate, 29 l
Bowers, N.R., 41,104, 105, 272
Breslow, N. 133
Brofitt, J.D., 133, 143, 146, 319
Brown, K.S., 216
Brown, R.L.,xv, 8, 216
Bureau of Labor Statistics, 273
Cantelli, F.P., 133
Censored data, I 0, 277
left-censored, 268
right-censored, 268
Central rate, 28,49
estimate, 117
exponential distribution, 62
relationship with probability, 53
uniform distribution, 60
Chiang, C.L., xii
Chi-square distribution, 200-20 l, 303
Cohort, 9,79
Complete data, 7,79
Composite hazard rate model, 297-303
Concomitant variables,5, 206-215, 280
Conditional proababilities, 22, 46
exponential distribution, 61
hyperbolic distribution, 64
relationship with central rate, 53
select, 23, 67
uniform distribution, 59
use in estimation, 29,89, 99,134
Consistency, 167, 170,321
Constant prepayment rate model
(CPR),285-289
Continuing care retirement communi­
ties, 258-265
Continuous-time st-0chastic process,
263
Controlled-data studies, IO
Corporate bond default rates, 289-291
Covariance, 34, 35
estimated multinomial
probabilities, 94
estimated survival function, 94
multinomial variables, 87
Cowell,M., 255
Cox, D.R.,214
358
Cox model, 214, 280
Cross-sectional studies, 9
Crowley,J., 133
Cumulative distribution function, 13
lower truncated, 24
multi-state model, 254
transformed variable, 30,32
upper and lower truncated, 26
Cumulative hazard function, 16,275,
300
estimation of, 172-173
Cumulative mortality rate, 291
Current Population Survey, 268
Curtailments, 282
Darling,D.A., 205
Decimal years, 226-227
Default loss rate measurement, 290
Default rate measurement, 289-290
Devine, T.J.; 267
Discrete-time stochastic process, 263
Distribution assumptions, 7
exponential, 61-63, l 27-129, 149,
152,156, 157, 161,163,171, 197,.
250,261,270,278
hyperbolic, 63-65, l 18, l 97
uniform, 59-61,119,125-127,149,
152-154,155,157,161,163,171,
197
Distribution of estimators, 322,324
Dobson,R.H.,40
Double-decrement environment,
density functions, 105,106
moment estimation, l 24-130
parametric estimation, 195-196
probabilities, 104-110
Drolette,M.E., 141,177
Duration
dependence, 269-270
of economic events, 267-280
of unemployment, 267-268
transfonnation of, 271
vectors, 228, 232
Index
Econometric prepayment model,
288-289
Economic theory, 272
Efficiency, 167, 322
Efron, 8., 312
Elandt-Johnson, R.C., xii, 7
Empirical (observed) survival
distribution, 81, 186,30 l
expected value, 84
variance, 84
Enders (see scheduled ending age)
Estimation, 7-8
actuarial (see Actuarial estimation
empirical survival distribution, 8 l
hazard rate, 90-91
maximum likelihood (see Maximum
likelihood estimation)
method of moments (see Moment
estimation)
minimum x2 , 20 l
minimum modified x 2 , 20 l
multinomial model, 85-9 l
parametric models, 181
use of conditional probabilities,29
89,99,134
Expectation of life, 17, 45
conditional, 47
curtate, 53
uniform distribution, 60
Exponential distribution (see Distribu­
tion assumptions, exponential)
Exposure, 50-53, l 16-118, 314-315
actuarial, 131-133, 138,230
exact, 117, l 38, 149, l 52, 229
scheduled, l l 6-117, 138, 230
Fabozzi, J.J., 282, 283
Failure mass, 312
Fannie Mae, 283
Federal Housing Administration
(FHA), 281
FHA prepayment model, 287
Fiscal ages, 225, 239-242
Fiscal year of birth, 239
Index
Follow-up time, 80
Force of disability, 247
Force of progression, 249
Force of transfer, 246
Force of transition, 260
Force of mortality, 15, 17, 43-44
estimated, 197-200
exponential distribution, 61-62
hyperbolic distribution, 64
select,17, 208-209
uniform distribution, 59
Frankfurt study, 255
Freddie Mac, 283
Full data,145,151-155
Future lifetime random variable, 3,5
distribution, 13
expected value, 16, 17,27,45
median, 17
truncated distribution, 24-26
variance, 16,27,46,48
Gehan,E.A., 198,305
Geometric distribution, 269
Gershenson,H.,xii, 132,133
Ginnie Mae, 283
Gompertz,B., 20
Graduation, xii,83
Greenwood,M.,136
Greenwood's formula, 136, 144, 170,
273,275
Grouping, 102
age last birthday, 231,240
calendar age, 231,235,240
maximum likelihood
estimation, 153-157
moment estimation, 118-119
Hazard rate function, 15-16
double-decrement (uniform), 125
estimated, 90-91,192
proportional model, 221
transformed variable, 31,32
upper and lower truncated, 26
Hickman,W.B., 289
Hoem,J.M., 129,133
359
Hoem's moment estimation, 129-130
Hogg, R.V., 21
Hollander, M.,202
Hoskins,W., 255
Hyperbolic distribution (see Distribu­
tion assumptions, hyperbolic)
Hypothesis testing,
Anderson-Darling, 205-206
chi-square, 200�20 l
Kolmogorov-Smimov, 202
modifiedA 2 , 206
modified K-S, 204-205
Incomplete data, 10
Independence assumption,105,161,
164
Index of industrial production, 280
Indicator variables, 150,151,161,
194,195,300
Insuring ages,225,233-238
Johnson,N. L., xii,7
Jones,B.L.,258
Jordan, C.W., 41,70,104,247
Kalbfleisch, J.D., xii
Kaplan, E.L.,167
Kaplan-Meier estimator (see productlimit estimator)
Kellison, S.G. , 281,189
Kennan, J ., 273
Keyfitz, N.,xii
Kiefer, N.M., 267
Klugman, S.A., 21, 143
Kolmogorov-Smimov statistic,
202-205
Lawless,J.F., xii,306
Lazar,P., 123
Least-squares,186-190, 191,197-200,
208-209, 306
Lee,E.T., xii
Length-biased sampling, 268
Index
360
Life tables, 7, 39
detenninistic interpretation, 41
method of estimation, 313
probabilistic interpretation, 41
radix, 40, 283, 315
select, 67-70
traditional form, 40
Linear-exponential model, 212
Liquidity, 282
Loan-to-value ratio, 281
Log-likelihood, 147
Log-linear exponential model, 214
Log-logistic distribution, 271
Log transformation, 197, 199, 208, 209
London, D., 83
Longitudinal studies, 9
Lymphadenopathy syndrome
(LAS), 249
Makeham, W.M., 21
Marginal mortality rate, 291
Marginal probability, 159
Marshall, E.W., 133
Maximum likelihood estimation,
bias, 165, 324
concomitant variables, 211-215
double-decrement
full data (general), 160-162
partial data (Case A), 162-165
multi-state models, 256-257
parametric models, 182, 185, 191,
192-197, 277-279
piecewise-exponential, 296
Rayleigh distribtuion, 304
single-decrement
full data (Case A), 148-149
full data (general), 151-154
partial data (Case A), 146-147
partial data (Case C), 155-159
random censoring, 158-159
variance, 165-166, 182, 324-326
Mean square error, 320-321
Meier, P., 167
Mereu, J.A., 66
Migration, 9
Miller, R.G., xii
Minimum area criteria, 303
Moment estimation
bias, 119-121
double-decrement
exponential, 127-129
uniform, 125-127
exposure interpretation, 116-118
general single-decrement
estimator, 114
grouping, 118-119
parametric models, 182
Schwartz-Lazar estimator, 122-123
special case estimators, 114-116
variance, 121-122
Mortgage-backed securities (MBS),
281
agency, 283
nonagency, 283
Mortgage loan prepayments, 281-289
Mortgage originator, 282
Mortgage pass-through securities, 283
Multiplicative model, 214-215
Multi-state models, 247-265
Negative binomial distribution, 269
Negative duration dependence, 269
Nelson-Aalen estimator, 172-174
Nelson, W.A., 173
Nesselle, D., 199
Newton-Raphson procedure, 305
Observatation period, 9, 97-98,
231, 234-235, 240
Panjer, H.H., 248, 258, 263
Parametric estimation,
concomitant variables, 206-215
hypothesis testing, 200-206
least-squares, 186-190, 19 l ,
197-200, 208-209
maximum likelihood, 182, 185, 191,
192-197
medians, 183
moments, I 82, 183
observed survival distribution, 186
percentiles, 184
361
llldex
Partial data, 146
Payment-to-income ratio, 281
Piecewise-exponential model, 295-297
Poisson distribution, 297
Pollard, J.H., xii, 105, 313
Polynomial hazard rate models,
303-308
Positive duration dependence, 269
Prentice, R.L., xii
Prepayment
models, 282, 283-289
option, 282
risk, 282
Probability density function, 14, 44
conditional, 23, 47
exponential distribution, 62
hyperbolic distribution, 64
likelihood function, 148
lower truncated, 24-25
multi-state models, 254
piecewise-exponential model, 296
Rayleigh distribution, 304
transformed variable, 30, 32
uniform distribution, 59
upper and lower truncated, 26
Weibull distribution, 270
Product-limit estimator,
interval probability, 167-171
survival function, 171-172, 273,
308-313
variance, 170, 171-172
Proportional hazard rate model, 221
Public Securities Association model
(PSA), 288
Quadratic hazard rate model, 306-308
Random censoring, 158-159, 257
Rao, C.R., 323
Rayleigh distribution, 303
Redfield, R., 249
Redistribute-to-the-right, 312-313
Reservation wage, 272
Risk set, 169
Robinson, J.M., 164
Rohatgi, V.K., 323
Ross, S.M., 264
Scheduled ending age, 98, IOI, 129,
132, 227, 313
Schwartz, D., 123
Schwartz-Lazar estimator, 122-123
Seal, H.L., 132, 133
Secondary whole-loan market, 282
Select life table (see Life tables, select)
Shooman, M.L., 299
Siddiqui, M.M., 198
Simultaneous events, 169
Singh, A.K., 299
Single monthly mortality, 286-287
Special Case A, I 02
maximum likelihood estimator,
146-150, 162-165, 168
moment estimator, 114, 126, 128
Special Case B, I 02
bias in moment estimator, 120-121
general moment estimator, 115
grouped moment estimator, 118-119
maximum likelihood estimator, 153,
166
Special Case C, l 02
general moment estimator, 116
grouped moment estimator, 119
maximum likelihood estimator, 154,
155-159
Spiegelman, M., xii
Stephens, M.A., 204, 206
Stochastic process, 248, 262-265
Study design,
cross-sectional, 9, 97
longitudinal, 9, 79, 193
truncated, I0, 193
Survival distribution function, 13
estimation of, 134-137, 171-172
lower truncated, 24
multi-state models, 254
transformed variable, 30, 32
upper and lower truncated, 25
362
Survival models,
actuarial, 4, 17
aggregate, 5
chi-square, 37
composite hazard rate, 297-303
concomitant variables, 206-215
exponential, 19, 28, 31, 32, 182,
183, 188, 191, 193, 194,204
gamma, 184
Gompertz, 20, 184, 197-199,221
log-logistic, 271
lognonnal, 38
Makeham, 21, 199-200
multinomial, 86
parametric, 6, 181-216
piecewise-exponential, 295-297
Poisson, 297
Rayleigh, 303
select, 5, 207-208
tabular, 6
Type I Least Value, 37
uniform, 18, 192, 202-204
Weibull, 21, 185, 191, 199, 209,
270-271, 278-279
Taylor series, 33, 34, 199
T-date, 239
Tenebein, A., 7, 207
Transformations,
decreasing, 30-31
increasing, 29-30
linear, 34
mean, 33-35
of duration, 271
variance, 33-35
Index
Truncated data, l O
Twelve-year prepaid life model, 284
Two-state models, 247
Unemployment,duration of, 267-268
Uniform distribution (see Distribution
assumptions, uniform)
Utility theory, 272
Valuation year of birth, 233
Vanderhoof, I., 7, 207
Variance, 320
actuarial estimators, 13 3
binomial proportion, 88
empirical survival distribution, 84
estimated hazard rate, 91
estimated multinomial
probability, 87
estimated survival function, 135-137
forces of transition, 258
future lifetime, 16, 27, 46, 48
maximum likelihood estimators,
165-166,324-326
multinomial variable, 86
product limit estimator, 170,
171-172
relation to mean square error,
320-321
Veterans Administration (VA), 281
Walter Reed Staging Method, 249
Willmot, G.E., 263
Wolfe, D.A., 202
Wolfenden, H.H., 133