The Conway-Maxwell-Poisson distribution: distributional theory and approximation

TL;DR

This paper explores the properties, convergence, and approximation methods for the Conway-Maxwell-Poisson (CMP) distribution and its binomial analogue, providing theoretical insights and bounds relevant for flexible statistical modeling.

Contribution

It establishes properties of CMP and CMB distributions, introduces convergence results, and derives bounds on their total variation distance, advancing understanding of these distributions.

Findings

Properties of CMP and CMB distributions are characterized.

Convergence results between CMB and CMP are established.

A bound on total variation distance is derived.

Abstract

The Conway-Maxwell-Poisson (CMP) distribution is a natural two-parameter generalisation of the Poisson distribution which has received some attention in the statistics literature in recent years by offering flexible generalisations of some well-known models. In this work, we begin by establishing some properties of both the CMP distribution and an analogous generalisation of the binomial distribution, which we refer to as the CMB distribution. We also consider some convergence results and approximations, including a bound on the total variation distance between a CMB distribution and the corresponding CMP limit.