Abstract
The topic of the present chapter is recursions for compound mixed Poisson distributions, in particular mixed Poisson distributions. As a simple motivating example, we consider a Gamma mixing distribution. Special attention is given to finite mixtures. We mainly concentrate on the Willmot class of continuous mixing distributions where the derivative of the natural logarithm of the Laplace transform of the mixing distribution is expressed as the ratio between two polynomials. In particular, we discuss some invariance properties of distributions in the Willmot class. Finally, we look at some specific parametric classes of mixing distributions within the Willmot class.
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