Characterization of count data distributions involving additivity and binomial subsampling

TL;DR

This paper characterizes all count data distributions with multiple parameters that are closed under addition and binomial subsampling, revealing that only certain Hermite families satisfy these properties, including Poisson and Hermite distributions.

Contribution

It provides a complete characterization of count distribution families satisfying additivity and binomial subsampling closure, identifying Hermite distributions as the exclusive solutions.

Findings

Poisson distribution is the case r=1.

Hermite distributions are the case r=2.

Few families satisfy both properties simultaneously.

Abstract

In this paper we characterize all the $r$-parameter families of count distributions (satisfying mild conditions) that are closed under addition and under binomial subsampling. Surprisingly, few families satisfy both properties and the resulting models consist of the $r$th-order univariate Hermite distributions. Among these, we find the Poisson ($r=1$) and the ordinary Hermite distributions ($r=2$).