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Transcript of A Primer on the Exponential Family of Distributions David Clark & Charles Thayer American...
![Page 1: A Primer on the Exponential Family of Distributions David Clark & Charles Thayer American Re-Insurance GLM Call Paper - 2004.](https://reader034.fdocuments.net/reader034/viewer/2022051401/56649de85503460f94ae2138/html5/thumbnails/1.jpg)
A Primer on the Exponential Family of Distributions
A Primer on the Exponential Family of Distributions
David Clark & Charles Thayer
American Re-Insurance
GLM Call Paper - 2004
![Page 2: A Primer on the Exponential Family of Distributions David Clark & Charles Thayer American Re-Insurance GLM Call Paper - 2004.](https://reader034.fdocuments.net/reader034/viewer/2022051401/56649de85503460f94ae2138/html5/thumbnails/2.jpg)
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AgendaAgenda
• Brief Introduction to GLM
• Overview of the Exponential Family
• Some Specific Distributions
• Suggestions for Insurance Applications
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Context for GLMContext for GLM
Linear Regression
Generalized Linear Models
Maximum Likelihood
XYE ][
Y~ Normal
XhYE ][
Y ~ Exponential Family
,][ XhYE
Y ~ Any Distribution
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Advantages over Linear Regression
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
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Transforming the Response versus
Transforming the Covariates
Transforming the Response versus
Transforming the Covariates
Linear Regression GLM
E[g(y)] = X· E[y] = g-1(X·)
Note that if g(y)=ln(y), then Linear Regression cannot handle any points where y0.
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Advantages of this Special Case of Maximum LikelihoodAdvantages 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
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Advantages of this Special Case of Maximum LikelihoodAdvantages 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
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General Form of the Exponential FamilyGeneral Form of the Exponential Family
iiiiii yhgyedyf exp ;
Note that yi can be transformed with any function e().
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“Natural” Form of the Exponential Family
“Natural” Form of the Exponential Family
,exp , ; iiii
ii yca
byyf
Note that yi is no longer within a function. That is, e(yi)=yi.
![Page 10: A Primer on the Exponential Family of Distributions David Clark & Charles Thayer American Re-Insurance GLM Call Paper - 2004.](https://reader034.fdocuments.net/reader034/viewer/2022051401/56649de85503460f94ae2138/html5/thumbnails/10.jpg)
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Specific Members of the Exponential Family
Specific Members of the Exponential Family
• Normal (Gaussian)
• Poisson
• Negative Binomial
• Gamma
• Inverse Gaussian
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Some Other Members of the Exponential Family
Some Other Members of the Exponential Family
• Natural FormBinomialLogarithmicCompound Poisson/Gamma (Tweedie)
• General Form [use ln(y) instead of y]LognormalSingle Parameter Pareto
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Normal DistributionNormal Distribution
Natural Form:
2ln
2
12/exp)(
22 y
yyf
The dispersion parameter, , is replaced with 2 in the more familiar form of the Normal Distribution.
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Poisson DistributionPoisson Distribution
Natural Form:
))!/ln((
ln)ln(exp)(Prob
yyy
yY
“Over-dispersed” Poisson allows 1.
Variance/Mean ratio =
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Negative Binomial DistributionNegative Binomial Distribution
Natural Form:
/
1ln/lnlnexp)(Prob
)(
yk
k
ky
kyY
yk
The parameter k must be selected by the user of the model.
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Gamma DistributionGamma Distribution
Natural Form:
)(ln)ln()1()ln(exp)(
y
yyf
Constant Coefficient of Variation (CV):
CV = -1/2
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Inverse Gaussian DistributionInverse Gaussian Distribution
Natural Form:
3
22ln
2
111
2exp)( y
y
yyf
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Table of Variance FunctionsTable 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
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The Unit Variance FunctionThe Unit Variance Function
We define the “Unit Variance” function as
V() = Var(y) / a()
That is, =1 in the previous table.
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Uniqueness PropertyUniqueness Property
The unit variance function V() uniquely identifies its parent distribution type within the natural exponential family.
f(y) V()
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Table of Skewness CoefficientsTable of Skewness Coefficients
Distribution Skewness
Normal 0
Poisson CV
Negative Binomial [1+/(+k)]·CV
Gamma 2·CV
Inverse Gaussian 3·CV
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Graph of Skewness versus CVGraph of Skewness versus CV
0
1
2
3
4
5
6
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)
Co
effi
cien
t o
f S
kew
nes
s
NegativeBinomial
LogNormal
InverseGaussianGamma
Poisson
Normal
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The Big Question:The Big Question:
What should the variance function look like for insurance applications?
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What is the Response Variable?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)
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An Example for Considering Variance Structure
An Example for Considering Variance Structure
Accident OnLevel Trended LossYear Premium Ult. Loss Ratio
1994 290,662 1,275,543 438.84%1995 391,490 47,490 12.13%1996 72,742,613 70,544,925 96.98%1997 265,124,454 161,625,762 60.96%1998 279,159,910 173,569,322 62.18%1999 339,612,341 246,497,223 72.58%2000 439,322,504 290,588,625 66.14%2001 469,582,172 327,742,407 69.79%2002 524,216,086 312,057,030 59.53%2003 869,036,055 689,968,152 79.39%
How would you calculate the mean and variance in these loss ratios?
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Defining a Variance StructureDefining a Variance Structure
We intuitively know that variance changes with loss volume – but how?
This is the same as asking
“V() = ?”
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Defining a Variance StructureDefining 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 1<p<2
• Weighted L-S Var(y) = /w
![Page 27: A Primer on the Exponential Family of Distributions David Clark & Charles Thayer American Re-Insurance GLM Call Paper - 2004.](https://reader034.fdocuments.net/reader034/viewer/2022051401/56649de85503460f94ae2138/html5/thumbnails/27.jpg)
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The Negative BinomialThe Negative Binomial
The variance function:
Var(y) = · + (/k)·2
random systematic
variance variance
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The “Tweedie” DistributionThe “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.
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Defining a Variance StructureDefining a Variance Structure
Negative Binomial
Tweedie
kCV
lim
0lim
CV
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Defining a Variance StructureDefining a Variance StructureComparison of Negative Binomial and Tweedie CV's
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
100 1,000 10,000 100,000
Logarithm of Expected Loss Size
Co
effi
cien
t o
f V
aria
tio
n (
CV
)
Negative Binomial Tweedie (p=1.5)
Asymptotic to .200
Asymptotic to 0
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Weighted Least-SquaresWeighted 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 least-squares: L-S = Σ(yi-i)2·wi
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ConclusionConclusion
A model fitted to insurance data should reflect the variance structure of the phenomenon being modeled.
GLM provides a flexible tool for doing this.