Families of power series distributions, with particular reference to the Lerch family

Abstract

The paper revisits the concept of a power series distribution by defining its series function, its power parameter, and hence its probability generating function. Realization that the series function for a particular distribution is a special case of a recognized mathematical function enables distributions to be classified into families. Examples are the generalized hypergeometric family and the q-series family, both of which contain generalizations of the geometric distribution. The Lerch function (a third generalization of the geometric series) is the series function for the Lerch family. A list of distributions belonging to the Lerch family is provided. The advantage of classifying power series distributions into families is that distributions within a family can be expected to have analogous properties which depend on the mathematical properties of the series function. This is demonstrated by focussing on equilibrium birth and death processes for the Lerch family.