EM Algorithm for Mixed Poisson and Other Discrete Distributions

Mixed Poisson distributions are widely used in various disciplines including actuarial applications. The family of mixed Poisson distributions contains several members according to the choice of the mixing distribution for the parameter of the Poisson distribution. Very few of them have been studied in depth, mainly because of algebraic intractability. In this paper we will describe an EM type algorithm for maximum likelihood estimation for mixed Poisson distributions. The main achievement is that it reduces the problem of estimation of the mising distribution which is usually easier. Variants of the algorithm work even when the probabiltity function of the mixed distribution is not nknown explicitly but we have only an approximation of it. Other discrete distributions are treated as well.