On the Compound Generalized Poisson Distributions

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Abstract
Goovaerts and Kaas (1991) present a recursive scheme, involving Panjer's recursion, to compute the compound generalized Poisson distribution (CGPD). In the present paper, we study the CGPD in detail. First, we express the generating functions in terms of Lambert's W function. An integral equation is derived for the pdf of CGPD, when the claim severities are absolutely continuous, from the basic principles. Also we derive the asymptotic formula for CGPD when the distribution of claim severity satisfies certain conditions. Then we present a recursive formula somewhat different and easier to implement than the recursive scheme of Goovaerts and Kaas (1991), when the distribution of claim severity follows an arithmetic distribution, which can be used to evaluate the CGPD. We illustrate the usage of this formula with a numerical example.