Claim dependence in hierarchical credibility models and estimation of structural parameters - Ebrahimzadeh, Mahdi (2012)
One of the most important techniques used in general insurance pricing is the credibility ratemaking. In general we can say, credibility theory is a quantitative tool that allows an insurer to combine the past experience of a policyholder to the premium in a risk class or group of risk classes. In the usual credibility model, observations are made of a risk or group of risks selected from a population and claims are assumed to be independent between different risks. However, there are some problems in practical applications and it may be violated in some situations. Some credibility models typically allow for one source of claim dependence only that is across time for an individual insured risk or a group of homogeneous insured risks. There exist some other credibility models in the literature which have been developed on two-level common effects model that allows for two possible sources of dependence: across time for the same individual risk and that between individual risks. In this thesis, we established the notion of modeling claim dependence in credibility models with dependence induced by three-level common effects that allows for three possible sources of dependence: the dependence among portfolio risks, dependence of the individual risks and the dependence of experience for a particular individual risk over time. Using conditional expectation, the credibility premium formulas in which the common effects random variables have a normal distribution are calculated and we present some obvious asymptotic properties of the credibility premium formula. We further give illustrative example to demonstrate the ideas. We also obtain the corresponding credibility formulas for the general (distribution-free) hierarchical structure credibility premiums in the model with three-level of common effects by using the projection method. Then we derive the general hierarchical structure or multi-level credibility premiums for the models with h-level of common effects. We also estimate the structural parameters of credibility models with dependence induced by common effects. The main advantage of our estimators is their simplicity in calculation and application. We derive unbiased estimators of structural parameters for two- and three-level common-effect models for portfolios with the Bühlmann model's structure. The results are extended to the Bühlmann-Straub model. We conjecture the hlevel model formulas when all assumptions are maintained. To illustrate numerically the three-level common effects model, claims data are generated. The result showed that the differences between the true values and our unbiased estimators are generally rather small. Lastly we illustrate the application of our model using real data.The result showed that the three-level model is better than two- and one-level common effects models. Furthermore, three-level model has the advantage of determining the influence of common effects at each level.