The (a, b, r) class of discrete distributions with applications

Abstract

In the insurance field the number of events such as losses to the insured or claims to the insurance company are an important aspect of loss modeling. Understanding the size of claims in terms of numbers and amounts makes it possible to modify and address issues related to creating insurance contracts. In general, certain counting (or discrete) distributions are used to model the number and amount of claims. There are situations where the modelled probability of having no claim is high. Indeed this is a desirable case for the benefit of insurance companies. An approach in modeling the number of claims in this case is by using Panjer’s (a, b, 1) class of discrete distributions. In this thesis, we look at a more general case of this class of distributions where there is an excess of claims at 0 to say r. We modify the existing (a, b, 1) model by assigning values greater than 0 to p0 (the probability of no claims) all the way up to pr (the probability of r claims). We then analyze this new model in terms of goodness of fit to actual claim data and compare with the classical (a, b, 1) and (a, b, 0) class of discrete distributions. This is done by using the maximum likelihood estimate (MLE) in estimating the parameters of each distribution discussed. In addition, the Akaike information criterion (AIC) is used to choose between competing distributions. This new model will be called (a, b, r) class of distributions, where r > 1.