Mixed Compound Poisson Distributions

The distribution of total claims payable by an insurer is considered when the frequency of claims is a mixed Poisson random variable. It is shown how in many cases the total claims density can be evaluated numerically using simple recursive formulae (discrete or continuous). Mixed Poisson distributions often have desirable properties for modelling claim frequencies. For example, they often have thick tails which make them useful for long-tailed data. Also, they may be interpreted as having arisen from a stochastic process. Mixing distributions considered include the inverse Gaussian, beta, uniform, non-central chi-squared, and the generalized inverse Gaussian as well as other more general distributions. It is also shown how these results may be used to derive computational formulae for the total claims density when the frequency distribution is either from the Neyman class of contagious distributions, or a class of negative binomial mixtures. Also, a computational formula is derived for the probability distribution of the number in the system for the M/G/1 queue with bulk arrivals.