A Primer on the Exponential Family of Distributions David Clark & Charles Thayer American Re-Insurance GLM Call Paper

2 Agenda Brief Introduction to GLM Overview of the Exponential Family Some Specific Distributions Suggestions for Insurance Applications

3 Context for GLM Linear Regression Generalized Linear Models Maximum Likelihood Y~ NormalY ~ Exponential FamilyY ~ Any Distribution

4 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

5 Transforming the Response versus Transforming the Covariates Linear RegressionGLM E[g(y)] = X·  E[y] = g -1 (X·  ) Note that if g(y)=ln(y), then Linear Regression cannot handle any points where y  0.

6 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

7 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

8 General Form of the Exponential Family Note that y i can be transformed with any function e().

9 “Natural” Form of the Exponential Family Note that y i is no longer within a function. That is, e(y i )=y i.

10 Specific Members of the Exponential Family Normal (Gaussian) Poisson Negative Binomial Gamma Inverse Gaussian

11 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

12 Normal Distribution Natural Form: The dispersion parameter, , is replaced with  2 in the more familiar form of the Normal Distribution.

13 Poisson Distribution Natural Form: “Over-dispersed” Poisson allows   1. Variance/Mean ratio = 

14 Negative Binomial Distribution Natural Form: The parameter k must be selected by the user of the model.

15 Gamma Distribution Natural Form: Constant Coefficient of Variation (CV): CV =  -1/2

16 Inverse Gaussian Distribution Natural Form:

17 Table of Variance Functions DistributionVariance Function Normal Var(y) =  Poisson Var(y) =  ·  Negative Binomial Var(y) =  ·  +(  /k)·  2 Gamma Var(y) =  ·  2 Inverse Gaussian Var(y) =  ·  3

18 The Unit Variance Function We define the “Unit Variance” function as V(  ) = Var(y) / a(  ) That is,  =1 in the previous table.

19 Uniqueness Property The unit variance function V(  ) uniquely identifies its parent distribution type within the natural exponential family. f(y)  V(  )

20 Table of Skewness Coefficients DistributionSkewness Normal 0 Poisson CV Negative Binomial[1+  /(  +k)]·CV Gamma 2·CV Inverse Gaussian 3·CV

21 Graph of Skewness versus CV

22 The Big Question: What should the variance function look like for insurance applications?

23 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)

24 An Example for Considering Variance Structure How would you calculate the mean and variance in these loss ratios?

25 Defining a Variance Structure We intuitively know that variance changes with loss volume – but how? This is the same as asking “ V(  ) = ?”

26 Defining a Variance Structure We want CV to decrease with loss size, but not too quickly. GLM provides several approaches: Negative BinomialVar(y) =  ·  +(  /k)·  2 TweedieVar(y) =  ·  p 1<p<2 Weighted L-SVar(y) =  /w

27 The Negative Binomial The variance function: Var(y) =  ·  + (  /k)·  2 random systematic variance variance

28 The “Tweedie” Distribution TweedieNeg. Binomial FrequencyPoisson Poisson SeverityGammaLogarithmic (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.

29 Defining a Variance Structure Negative Binomial Tweedie

30 Defining a Variance Structure

31 Weighted Least-Squares Use Normal Distribution but set a(  ) =  /w i such that, variance is proportional to some external exposure weight w i. This is equivalent to weighted least- squares:L-S = Σ(y i -  i ) 2 ·w i

32 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.