Statistical distributions in general insurance stochastic processes

Abstract

A general insurance risk model consists of in initial reserve, the premiums collected, the return on investment of these premiums, the claims frequency and the claims sizes. Except for the initial reserve, these components are all stochastic. The assumption of the distributions of the claims sizes is an integral part of the model and can greatly in uence decisions on reinsurance agreements and ruin probabilities. An array of parametric distributions are available for use in describing the distribution of claims. The study is focussed on parametric distributions that have positive skewness and are de ned for positive real values. The main properties and parameterizations are studied for a number of distributions. Maximum likelihood estimation and method-of-moments estimation are considered as techniques for tting these distributions. Multivariate numerical maximum likelihood estimation algorithms are proposed together with discussions on the e ciency of each of the estimation algorithms based on simulation exercises. These discussions are accompanied with programs developed in SAS PROC IML that can be used to simulate from the various parametric distributions and to t these parametric distributions to observed data. The presence of heavy upper tails in the context of general insurance claims size distributions indicates that there exists a high risk of observing very large and even extreme claims. This needs to be allowed for in the modeling of claims. Methods used to describe tail weight together with techniques that can be used to detect the presence of heavy upper tails are studied. These methods are then applied to the parametric distributions to classify their tails' heaviness. The study is concluded with an application of the techniques developed to t the parametric distributions and to evaluate the tail heaviness of reallife claims data. The goodness-of- t of the various tted distributions are discussed. Based on the nal results further research topics are identi ed.