Mixed Poisson distributions

Abstract

One of the most important counting distributional classes in insurance modelling is the class of mixed Poisson distributions. A mixed Poisson distribution is often used to model the number of losses or claims arising from a group of risks where the risk level among the group retains heterogeneity which can not be classified by underwriting criteria. However, it may be reasonable to assume that the risk level follows a probability distribution, and given the risk level the number of losses follows a Poisson distribution. Thus, the number of losses follows a mixed Poisson distribution. Examples can be found in Klugman, Panjer and Willmot (1998). An excellent reference for mixed Poisson distributions is the book by Grandell (1997). In this chapter, we focus on relations between the tail of a mixed Poisson distribution and its mixing distribution. We begin with a representation of tail probabilities of the mixed Poisson distribution, and discuss ratios of these tail probabilities and their connection to the radius of convergence of its probability generating function. Special attention is given to the mixing distribution and its reliability classification. We derive reliability based bounds for the ratios of tail probabilities. In the final section, we present some asymptotic results for mixed Poisson distributions.