Goodness of fit for the generalized Poisson distribution based on the probability generating function

Abstract

The family of generalized Poisson (GP) distributions, which contain among many others as special cases the Compound Poisson and Katz distributions, is a flexible family of distributions for modelling count data. The probability generating function (PGF) of the GP is the unique PGF satisfying certain differential equation. Based on this property, a goodness-of-fit test for the family of GP distributions is proposed and studied. The test is proved to be consistent against fixed alternatives and its null distribution can be consistently approximated by a parametric bootstrap. The goodness of the bootstrap estimator and the power for finite sample sizes are numerically assessed.