Predictive Modeling Applications in Actuarial Science

Chapter 5 - Generalized Linear Models


Curtis Gary Dean | Ball State University

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Generalized linear models (GLMs) generalize linear regression in two important ways: (1) the response variable y can be linked to a linear function of predictor variables xj with a nonlinear link function and (2) the variance in the response variable y is not required to be constant across observations but can be a function of y's expected value. For example, if y represents the number of claims, then the variance in y may depend on the expected value of y as in a Poisson distribution. In linear regression the normal distribution plays a key role but with GLMs the response variable y can have a distribution in a linear exponential family. These include distributions important to actuaries: Poisson, binomial, normal, gamma, inverse Gaussian, and compound Poisson-gamma. Actuaries can model frequency, severity, and loss ratios with GLMs as well as probabilities of events such as customers renewing policies.

The likelihood function has a key role in GLMs. Maximum likelihood estimation replaces least squares in the estimation of model coefficients. The log-likelihood function is used to perform statistical tests.