Extended Linear Model

Study guides: Exam MAS-I, Exam MAS-II

Extended Linear Models is the name the CAS MAS-I outline gives its regression domain: ordinary linear regression and the models that extend it by relaxing its classical assumptions — allowing a non-Normal response distribution, a non-identity Link Function, and a more flexible linear predictor. Generalized Linear Models are the most common extended linear model used in actuarial work.

\eta_i = \beta_0 + \beta_1 x_{i1} + \cdots + \beta_p x_{ip}

g(\mu_i) = \eta_i, \qquad Y_i \sim \text{(chosen response distribution)}

• Both classes can include interactions, transformations such as polynomial terms, and banded variables: "linear" means linear in the coefficients, so \beta_0 + \beta_1 x + \beta_2 x^2 is still a linear model
• Building an extended linear model means choosing its Model Structure, which has three parts: the response distribution, the Link Function, and the linear predictor (which predictors, transformations, and interactions enter it). The MAS-I outline's own example of choosing a GLM's model structure is choosing its link function and the distribution of the dependent variable
• Ordinary least-squares regression is a special case where the response is Normal and the link is the identity function
• Model adequacy is compared across candidate extended linear models using Deviance, AIC, BIC, and Residual Plots
• In insurance ratemaking, the choice of extended linear model is driven by the data's behavior — e.g., a right-skewed, strictly positive severity favors a Gamma distribution with a log link, while integer claim counts favor a Poisson distribution

Selecting an Extended Linear Model for Claim Severity An actuary has individual claim severities that are strictly positive and right-skewed. Why is ordinary least-squares regression not an appropriate extended linear model here, and what would be a better choice?

Answer Ordinary least-squares assumes a Normal response, which can produce negative fitted severities and underweights the right skew. A Gamma Generalized Linear Model with a log Link Function is a better extended linear model: it restricts fitted values to be positive and models the multiplicative, right-skewed behavior typical of claim severity.