Generalized Linear Models: Poisson and Gamma Regression

GLM Framework

A generalized linear model has three components: (1) a random component specifying the response distribution (from the exponential family), (2) a systematic component (the linear predictor eta = X*beta), and (3) a link function connecting the mean to the linear predictor: g(mu) = eta. Linear regression is a GLM with normal response and identity link. Logistic regression uses a Bernoulli response and logit link.

Parameters are estimated by maximum likelihood. The deviance measures goodness of fit, analogous to SSE in linear regression. AIC and BIC can be used for model comparison.

Poisson Regression

For count data (claim frequency), use a Poisson GLM with log link: ln(mu) = beta_0 + beta_1*X1 + ... + beta_p*Xp, so mu = exp(eta). Interpretation: a one-unit increase in Xj multiplies the expected count by exp(beta_j). An offset term ln(exposure) can be included when observations have different exposure periods, making the model rate-based rather than count-based.

The Poisson model assumes the mean equals the variance. If the data shows overdispersion (variance > mean), consider a negative binomial model or a quasi-Poisson approach.

Gamma Regression

For positive continuous data (claim severity), a Gamma GLM with log link is common: ln(mu) = eta. The gamma distribution allows variance proportional to the square of the mean (constant coefficient of variation), which fits many insurance severity distributions. The inverse link g(mu) = 1/mu is the canonical link for the gamma, but the log link is more commonly used in practice because of its interpretability. For Exam SRM, understand the choice of distribution and link function and how they relate to the data characteristics.