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GLM I: Introduction to Generalized Linear Models

GLM I: Introduction to Generalized Linear Models. By Curtis Gary Dean Vice President and Actuary,. Classical Multiple Linear Regression. Y i = a 0 + a 1 X i 1 + a 2 X i 2 …+ a m X im + e i Y i are the response variables and X ij are predictors

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GLM I: Introduction to Generalized Linear Models

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  1. GLM I: Introduction to Generalized Linear Models By Curtis Gary Dean Vice President and Actuary,

  2. Classical Multiple Linear Regression • Yi= a0 + a1Xi1 + a2Xi2 …+ amXim + ei • Yi are the response variables and Xij are predictors • ei is random error term: Normally distributed • E[ei] = 0 and Var(ei) = σ2 is constant

  3. Classical Multiple Linear Regression • μi = E[Yi] = a0 + a1Xi1 + …+ amXim • Yi is Normally distributed random variable with variance σ2 • Want to estimate μi = E[Yi] for each i

  4. Problems with Traditional Model • Number of claims is discrete • Claim sizes are skewed to the right • Probability of an event is in [0,1] • Variance is not constant across data points i • Nonlinear relationship between X’s and Y’s

  5. Generalized Linear Models - GLMs • Fewer restrictions • Y can model number of claims, probability of renewing, loss severity, loss ratio, etc. • Large and small policies can be put into one model • Y can be nonlinear function of X’s • Classical linear regression model is a special case

  6. Generalized Linear Models - GLMs • g(μi )= a0 + a1Xi1 + …+ amXim • g( ) is the link function • E[Yi] = μi= g-1(a0 + a1Xi1 + …+ amXim) • Yi can be Normal, Poisson, Gamma, Binomial, Compound Poisson, … • Variance can be modeled

  7. Exponential Family of Distributions

  8. Normal Distribution in Exponential Family

  9. Some Math Rules: Refresher

  10. Poisson Distribution in Exponential Family

  11. Poisson Distribution in Exponential Family

  12. Compound Poisson Distribution • Y = C1 + C2 + . . . + CN • N is Poisson random variable • Ciare i.i.d. with Gamma distribution • This is an example of a Tweedie distribution • Y is member of Exponential Family

  13. Members of the Exponential Family • Normal • Poisson • Binomial • Gamma • Inverse Gaussian • Compound Poisson

  14. Variance Structure • E[Yi] = μi = b' (θi) → θi= b' (-1)(μi ) • Var[Yi] = a(Φi) b' ' (θi) = a(Φi) V(μi) • Common form: Var[Yi] = Φ V(μi)/wi • Φis constant across data butweights applied to data points

  15. Variance Functions V(μ) V(μ) • Normal 0 • Poisson  • Binomial (1-) • Tweedie p, 1<p<2 • Gamma 2 • Inverse Gaussian 3 • Recall: Var[Yi] = Φ V(μi)/wi

  16. Variance at Point and Fit A Practioner’s Guide to Generalized Linear Models: A CAS Study Note

  17. Why Exponential Family? • Distributions in Exponential Family can model a variety of problems • Standard algorithm for finding coefficients a0, a1, …, am

  18. Modeling Number of Claims

  19. Assume a Multiplicative Model • μi = expected number of claims in five years • μi = BF,01 x CSex(i) x CTerr(i) • If i is Female and Terr 01 → μi = BF,01 x 1.00 x 1.00

  20. Multiplicative Model • μi = exp(a0 + aSXS(i) + aTXT(i)) • μi = exp(a0) x exp(aSXS(i)) x exp( aTXT(i)) • i is Female → XS(i) = 0; Male →XS(i) = 1 • i is Terr 01 → XT(i) = 0; Terr 02 →XT(i) = 1

  21. Natural Log Link Function • ln(μi) = a0 + aSXS(i) + aTXT(i) • μi is in (0 , ∞ ) • ln(μi) is in ( - ∞ , ∞ )

  22. Poisson Distribution in Exponential Family

  23. Natural Log is Canonical Link for Poisson • θi = ln(μi) • θi= a0 + aSXS(i) + aTXT(i)

  24. Estimating Coefficients a1, a2, .., am • Classical linear regression uses least squares • GLMs use Maximum Likelihood Method

  25. Likelihood and Log Likelihood

  26. Find a0, aS, and aT for Poisson

  27. Iterative Numerical Procedure to Find ai’s • Use statistical package or actuarial software • Specify link function and distribution type

  28. Solution to Our Example • a0 = -.288 → exp(-.288) = .75 • aS = .262 → exp(.262) = 1.3 • aT = .095 → exp(.095) = 1.1 • μi = exp(a0) x exp(aSXS(i)) x exp( aTXT(i)) • μi = .75 x 1.3XS(i) x 1.1XT(i) • i is Male,Terr 01 → μi = .75 x 1.31 x 1.10

  29. Other Common Link Functions • Identity link: canonical link for Normal • Logit link ln[ μ/ (1- μ) ]: canonical link for Binomial • maps (0,1) onto ( -∞, ∞) • Reciprocal 1/μ: canonical link for Gamma • But, you do not have to match link and distribution. Choose link and distribution that make sense!

  30. Measuring Goodness of Fit • Deviance is measure of goodness of fit • Compare different models • Compare effect of dropping/adding Xij’s • Also called log likelihood ratio statistic

  31. More on Deviance

  32. Generalized Linear Models - GLMs • Response Y can be discrete or continuous • Y can be number of claims, probability of renewing, loss severity, loss ratio, etc. • Large and small policies can be put into one model. • Y can be nonlinear function of X’s • Measure goodness of fit using Deviance

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