Some characterizations and properties of COM-Poisson random variables

Abstract

This paper introduces some new characterizations of COM-Poisson random variable. First, it extends Moran-Chatterji characterization, and generalizes Rao-Rubin characterization of Poisson distribution to COM-Poisson distribution. Then, it defines the COM-type discrete r.v. X ν of the discrete random variable X. The probability mass function of X ν has a link to the Rényi entropy and Tsallis entropy of order ν of X. And then we can get the characterization of Stam inequality for COM-type discrete version Fisher information. By using the recurrence formula, the property that COM-Poisson random variables (ν = 1) is not closed under addition are obtained. Finally, under the property of "not closed under addition" of COM-Poisson random variables, a new characterization of Poisson distribution is found.

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“…Due to its smooth transition between over-dispersion and under-dispersion, it plays a significant role in modelling count data [Hilbe (2014)] and has an extraordinarily diverse range of applications, see [Sellers, Borle and Shmueli (2012)] for a brief survey. However, despite some initiatives [Daly and Gaunt (2016), Li, Zhang and He (2020)], there is disproportionately little advance in its analytical properties and an approximation theory based on this popular distribution. This note aims to explain the fundamental reason behind the unbalanced development.…”

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