Expansion in Bell polynomials of the distribution of the total claim amount with Weibull-distributed claim sizes

Abstract

The total claim amount for a fixed period of time is, by definition, a sum of a random number of claims of random size.In this paper we explore the probabilistic distribution of the total claim amount for claims following a Weibull distribution, one which may serve as a satisfactory model for both small and large claims As models for the number of claims we use the geometric, Poisson, logarithmic and negative binomial distributions.In all these cases the densities of the total claim amount are obtained via Laplace transform of a density function, an expansion in Bell polynomials of a convolution and a subsequent Laplace inversion. doi:10.1017/S1446181108000187