Abstract

Generalized linear models (GLMs) are gaining popularity as a statistical analysis method for insurance data. We study the theory and applications of GLMs in insurance. The first chapter gives an introduction of the theory of GLMs and generalized linear mixed models (GLMMs) as well as the bias correction for GLM estimators. It is shown that the maximum likelihood estimators (MLEs) of the parameters in GLMs are asymptotically normal and asymptotically unbiased. However, when the sample size $n$ or the total Fisher information is small, the MLEs can be biased. The bias is usually ignored in practice. However, in small or moderate--size portfolios, a bias correction can be appreciable.

For segmented portfolios, as in car insurance, the question of credibility arises naturally; how many observations are needed in a risk class before the GLM estimators can be considered credible? In this thesis we study the limited fluctuations credibility of the GLM
estimators as well as in the extended case of GLMMs. We show how credibility depends on the sample size, the distribution of covariates and the link function. We give a criteria for full credibility of the GLM estimators. This provides a mechanism to obtain confidence intervals for the GLM and GLMM estimators.

If the full credibility criteria cannot be satisfied, it is
interesting to calculate the partial credibility matrix and the GLM estimators. Here, for a general link function the credibility matrix is not known explicitly. Under certain assumptions, numerical methods are developed to compute the GLM estimators and the credibility matrix. For some specific link functions, further properties are developed. For instance, Hachemeister's credibility
regression model is one such special case of our model, where the link function is linear.

Loss reserving is a major challenge for casualty actuaries due to the frequently changing business environments. Recently, some aggregate loss reserving models have been extended to or developed by research actuaries within the framework of GLMs. In this thesis we establish a structural loss reserving model which combines the
exposure and loss emergence patterns and the loss development pattern, again within the framework of a GLM. Discounted loss reserves can also be obtained from this model.