Compound Mixed Multivariate Poisson Distributions

Abstract

The purpose of this chapter is to extend the theory of Chap. 3 to a class of compound mixed multivariate Poisson distributions. We first give a presentation of multivariate Poisson distributions. In particular, we emphasise that, whereas such a distribution is multivariate so that when compounding it, it becomes a compound distribution of Type 2, it can also be expressed as a compound distribution of Type 1, so that we can apply the theory of compound multivariate distributions of Type 1. When extending the class of counting distributions to mixed distributions, we restrict the class of mixing distributions in such a way that this aspect is preserved. Like in the univariate case, the Gamma mixing distribution is a rather simple case, so we warm up with that. Turning to compound distributions of Type 1, we first treat a general case before restricting to the Willmot class. Then we describe how the theory can be used to evaluate compound mixed multivariate Poisson distributions by using the Type 1 representation of such distributions. Finally, we consider some specific parametric classes of mixing distributions within the Willmot class.