Estimation, Evaluation, and Selection of Actuarial Models

5.3. THE GENERALIZED LINEAR MODEL 93
and the additional parameters must not depend on the mean. Let z be a vector of covariates for
an individual, let β be a vector of coefficients, and let η(µ) and c(y) be functions. The generalized
linear model then states that the random variable, X, has as its distribution function
where µ is such that η(µ) =c(β T z).
F (x|θ, β) =F (x|µ, θ)
The model indicates that the mean of an individual observation is related to the covariates
through a particular set of functions. Normally, these functions do not involve parameters, but
instead are used to provide a good fit or to ensure that only legitimate values of µ are encountered.
Example 5.11 Demonstrate that the ordinary linear regression model is a special case of the generalized
linear model.
For ordinary linear regression, X has a normal distribution with µ = µ and θ = σ. Bothη and
c are the identity function, resulting in µ = β T z. ¤
The model presented here is more general than the one usually used. Usually, only a few
distributions are allowed to be used for X. The reason is that for these distributions, people have
been able to develop the full set of regression tools, such as residual analysis. Because such analyses
are covered in a different part of Part 4, that restriction is not needed here.
For many of the distributions we have been using, the mean is not a parameter. However, it
could be. For example, we could parameterize the Pareto distribution by setting µ = θ/(α − 1) or
equivalently, replacing θ with µ(α − 1). The distribution function is now
· µ(α − 1)
¸α
F (x|µ, α) =1−
, µ > 0, α > 1.
µ(α − 1) + x
Note the restriction on α in the parameter space.