Characterizing Additively Closed Discrete Models by a Property of Their Maximum Likelihood Estimators, With an Application to Generalized Hermite Distributions

This article reports on two-parameter count distributions (satisfying very general conditions) that are closed under addition so that their maximum likelihood estimator (MLE) of the population mean is the sample mean. The most important of these in practice, the generalized Hermite distribution, is analyzed, and a necessary and sufficient condition is given to ensure that the MLE is the solution of likelihood equations. Score test to contrast the Poisson assumption is studied, and two examples of applications are given.