A Primer on the Exponential
Family of Distributions
David Clark & Charles Thayer
American Re-Insurance
GLM Call Paper - 2004
Agenda
• Brief Introduction to GLM
• Overview of the Exponential Family
• Some Specific Distributions
• Suggestions for Insurance
Applications
2
Context for GLM
Maximum Likelihood
Generalized Linear Models
Linear Regression
 
E[Y ]  X  
Y~ Normal
 
E[Y ]  hX   
E[Y ]  h X ,  
Y ~ Exponential Family Y ~ Any Distribution
3
Advantages over Linear
Regression
• Instead of linear combination of
covariates, we can use a function of
a linear combination of covariates
• Response variable stays in original
units
• Great flexibility in variance structure
4
Transforming the Response
versus
Transforming the Covariates
Linear Regression
E[g(y)] = X·
GLM
E[y] = g-1(X·)
Note that if g(y)=ln(y), then Linear
Regression cannot handle any points
where y0.
5
Advantages of this Special
Case of Maximum Likelihood
• Pre-programmed in many software
packages
• Direct calculation of standard errors
of key parameters
• Convenient separation of Mean
parameter from “nuisance”
parameters
6
Advantages of this Special
Case of Maximum Likelihood
• GLM useful when theory immature,
but experience gives clues about:
How mean response affected by
external influences, covariates
How variability relates to mean
Independence of observations
Skewness/symmetry of response
distribution
7
General Form of the
Exponential Family
f  yi ;  i   exp d  i   e yi   g  i   h yi 
Note that yi can be transformed with
any function e().
8
“Natural” Form of the
Exponential Family
  i  yi  b i 

f  yi ;  i ,    exp 
 c yi ,  
a 


Note that yi is no longer within a
function. That is, e(yi)=yi.
9
Specific Members of the
Exponential Family
• Normal (Gaussian)
• Poisson
• Negative Binomial
• Gamma
• Inverse Gaussian
10
Some Other Members of the
Exponential Family
• Natural Form
Binomial
Logarithmic
Compound Poisson/Gamma (Tweedie)
• General Form [use ln(y) instead of y]
Lognormal
Single Parameter Pareto
11
Normal Distribution
Natural Form:
2


1
y
2
f ( y )  exp    y   / 2   
 ln
  2






2 

The dispersion parameter, , is replaced with
2 in the more familiar form of the Normal
Distribution.
12
Poisson Distribution
Natural Form:
 ln(  )  y   

ln  
Prob(Y  y )  exp 
 y
 ln(( y /  )!)




“Over-dispersed” Poisson allows   1.
Variance/Mean ratio = 
13
Negative Binomial Distribution
Natural Form:
   
 ( k  y )  1
 k  

  y  ln 
  k  /   ln 
Prob(Y  y )  exp ln 
 y /  
    k 
 k 


The parameter k must be selected by the
user of the model.
14
Gamma Distribution
Natural Form:
  y 

  
  ln(  )   (  1)  ln(   y )  ln 

f ( y )  exp 
  
 ( ) 

Constant Coefficient of Variation (CV):
CV = -1/2
15
Inverse Gaussian Distribution
Natural Form:
  y   1  1  1
f ( y )  exp  2       
 ln
 2       2y



2 y 

3
16
Table of Variance Functions
Distribution
Variance Function
Normal
Var(y) = 
Poisson
Var(y) =  ·
Negative Binomial
Var(y) =  ·+(/k)·2
Gamma
Var(y) =  ·2
Inverse Gaussian
Var(y) =  ·3
17
The Unit Variance Function
We define the “Unit Variance” function as
V() = Var(y) / a()
That is,  =1 in the previous table.
18
Uniqueness Property
The unit variance function V()
uniquely identifies its parent
distribution type within the natural
exponential family.
f(y)  V()
19
Table of Skewness Coefficients
Distribution
Skewness
Normal
0
Poisson
CV
Negative Binomial
[1+/(+k)]·CV
Gamma
2·CV
Inverse Gaussian
3·CV
20
Coefficient of Skewness
Graph of Skewness versus CV
6
Negative
Binomial
5
4
LogNormal
3
Inverse
Gaussian
Gamma
2
Poisson
1
Normal
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
Coefficient of Variation (CV)
21
The Big Question:
What should the variance
function look like for insurance
applications?
22
What is the Response Variable?
• Number of Claims
• Frequency (# claims per unit of exposure)
• Severity
• Aggregate Loss Dollars
• Loss Ratio (Aggregate Loss / Premium)
• Loss Rate (Aggregate Loss per unit of
exposure)
23
An Example for Considering
Variance Structure
Accident
Year
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
OnLevel
Premium
Trended
Ult. Loss
Loss
Ratio
290,662
391,490
72,742,613
265,124,454
279,159,910
339,612,341
439,322,504
469,582,172
524,216,086
869,036,055
1,275,543
47,490
70,544,925
161,625,762
173,569,322
246,497,223
290,588,625
327,742,407
312,057,030
689,968,152
438.84%
12.13%
96.98%
60.96%
62.18%
72.58%
66.14%
69.79%
59.53%
79.39%
How would
you calculate
the mean and
variance in
these loss
ratios?
24
Defining a Variance Structure
We intuitively know that variance
changes with loss volume –
but how?
This is the same as asking
“V() = ?”
25
Defining a Variance Structure
We want CV to decrease with loss size, but
not too quickly. GLM provides several
approaches:
• Negative Binomial
Var(y) = · +(/k)·2
• Tweedie
Var(y) = ·p
• Weighted L-S
Var(y) = /w
1<p<2
26
The Negative Binomial
The variance function:
Var(y) = · + (/k)·2
random
systematic
variance
variance
27
The “Tweedie” Distribution
Tweedie
Neg. Binomial
Frequency
Poisson
Poisson
Severity
Gamma
Logarithmic
(exponential when p=1.5)
Both the Tweedie and the Negative Binomial can
be thought of as intermediate cases between
the Poisson and Gamma distributions.
28
Defining a Variance Structure
Negative Binomial
Tweedie
lim
CV

lim
CV
 0
 
 

k
29
Defining a Variance Structure
Comparison of Negative Binomial and Tweedie CV's
0.7
Coefficient of Variation (CV)
0.6
0.5
0.4
Asymptotic to .200
0.3
0.2
0.1
Asymptotic to 0
0
100
1,000
10,000
100,000
Logarithm of Expected Loss Size
Negative Binomial
Tweedie (p=1.5)
30
Weighted Least-Squares
Use Normal Distribution but set
a() =  /wi
such that, variance is proportional to
some external exposure weight wi.
This is equivalent to weighted leastsquares:
L-S = Σ(yi-i)2·wi
31
Conclusion
A model fitted to insurance data should
reflect the variance structure of the
phenomenon being modeled.
GLM provides a flexible tool for doing this.
32