Stat 643 Exam 1
October 29, 2002
Prof. Vardeman
1. a) Give an example where X is complete for ) - @ and not complete for ) - @w § @.
b) Give an example showing that Theorem 8 is not true without the domination hypothesis.
2. Suppose that ] has pdf 0 on Ð!ß_Ñ and conditional on ] œ C, F is Bernoulli :ÐCÑ, for :Ð † Ñ
an increasing function Ð!ß_ÑpÒ!ß "Ó. Let
\ œ F†]
(\ œ ] when F œ " and \ œ ! otherwise. \ is a censored version of ] , where the censoring
probability depends on the realized value of ] .) T \ , the distribution of \ , is then absolutely
continuous wrt $! € -, for $! a unit point mass at ! and - Lebesgue measure on Ð!ß_Ñ . An R-N
derivative is
' _ Ð" • :ÐCÑÑ0 ÐCÑ.C
1ÐBÑ œ œ !
:ÐBÑ0 ÐBÑ
if B œ !
if B ž !
a) In the context above, suppose that 0 is the ExpÐ" Ñ density
"
C
0 ÐCl" Ñ œ MÒC ž !Ó expÐ • Ñ
"
"
and that
:ÐCÑ œ œ
C
"
if C • "
if C "
i) Show how to calculate the Fisher information in \ about " at the value "! , M \ Ð"! Ñ.
(You don't have to actually complete the integration or differentiation here, but you must
set up an expression appropriate for this specific model.)
ii) Under what conditions would you expect this to be essentially the same as the Fisher
information in ] ?
b)
Now suppose that \" ß \# ß á ß \8 are iid as in part a) above.
i) Consider the estimator of " ,
! \3 MÒ\3 ž "Ó
8
s8 œ
"
3œ"
8
! MÒ\3 ž "Ó
•"
3œ"
s 8 is consistent for " . (It may be helpful to know the calculus fact that
Argue that "
C
' CexpÐ • " Ñ.C œ • Ò" C € " # ÓexpÐ • "C Ñ )
~
s 8 that can probably be shown to have
ii) Suggest a modification (say " 8 ) of "
~
È8ˆ" 8 • "! ‰ converging in distribution to NÐ!ß "ÎM\ Ð"! ÑÑ under "! . (Again, you don't
have to be explicit here by completing nasty calculus, but you must set up something
specific for this model.)
c) Now suppose that :ÐCÑ has a parameter ), and in fact that
:ÐCl)Ñ œ œ
C)
"
if C • "
if C "
so that the distribution of \ has parameter vector Ð" ,)Ñ - Ð!ß_Ñ # . You may henceforth use the
abbreviation
:! Ð" ,)Ñ œ (
_
!
"
"
Ð" • :ÐCl)ÑÑ0 ÐCl" Ñ.C œ ( Ð" • C) Ñ expa • CÎ" b.C
"
!
(don't try to evaluate this function of " and )). Let \" ß \# ß á ß \8 be iid TÐ\" ,)Ñ .
i) Write out the log-likelihood function here, P8 Ð" ,)Ñ.
ii) Find a low-dimensional sufficient statistic and argue that it is minimal sufficient.
The partial MathCad printout attached to this exam concerns the analysis of P8 Ð" ,)Ñ for a
particular sample of size 8 œ $!. Use the information on this printout to answer the following
questions.
s ‡8 ß s)‡8 Ñ the (joint) MLE of Ð" ,)Ñ, what is a sensible standard error (estimated
iii) For Ð"
s ‡8 ?
standard deviation) for "
iv) What are approximate 95% confidence limits for " derived from inversion of
likelihood ratio tests of H! À " œ "! ?
Suppose that one proposes a prior distribution for Ð" ,)Ñ with " µ ExpÐ"Ñ independent of ) with
"
pdf 2Ð)Ñ œ MÒexpÐ • &Ñ • ) • expÐ&ÑÓ "!)
(this is ln()) µ Uniform Ð • &ß &Ñ). For the particular
sample referred to above, there is a contour plot of
"!$! Œ
expaP8 Ð" ß )Ñ • " b
•
"!)
on the printout.
v) Discuss briefly how it appears that "classical" (likelihood-based) and Bayes inferences
for Ð" ,)Ñ will differ and say why these differences are consistent with the prior
distribution used here.