1. A Primer on the Exponential Family of Distributions David Clark & Charles Thayer American Re-Insurance GLM Call Paper - 2004

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

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

4. 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. 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. 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. 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. 8 General Form of the Exponential Family Note that y i can be transformed with any function e().

9. 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. 10 Specific Members of the Exponential Family Normal (Gaussian) Poisson Negative Binomial Gamma Inverse Gaussian

11. 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. 12 Normal Distribution Natural Form: The dispersion parameter, , is replaced with  2 in the more familiar form of the Normal Distribution.

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

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

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

16. 16 Inverse Gaussian Distribution Natural Form:

17. 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. 18 The Unit Variance Function We define the “Unit Variance” function as V(  ) = Var(y) / a(  ) That is,  =1 in the previous table.

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

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

21. 21 Graph of Skewness versus CV

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

23. 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. 24 An Example for Considering Variance Structure How would you calculate the mean and variance in these loss ratios?

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

26. 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. 27 The Negative Binomial The variance function: Var(y) =  ·  + (  /k)·  2 random systematic variance variance

28. 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. 29 Defining a Variance Structure Negative Binomial Tweedie

30. 30 Defining a Variance Structure

31. 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. 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.