## Abstract

The Poisson, geometric and Bernoulli distributions are special cases of a flexible count distribution, namely the Conway-Maxwell-Poisson (CMP) distribution – a two-parameter generalization of the Poisson distribution that can accommodate data over- or under-dispersion. This work further generalizes the ideas of the CMP distribution by considering sums of CMP random variables to establish a flexible class of distributions that encompasses the Poisson, negative binomial, and binomial distributions as special cases. This sum-of-Conway-Maxwell-Poissons (sCMP) class captures the CMP and its special cases, as well as the classical negative binomial and binomial distributions. Through simulated and real data examples, we demonstrate this model’s flexibility, encompassing several classical distributions as well as other count data distributions containing significant data dispersion.

## Highlights

• The Poisson distribution is one of the most popular discrete distributions, serving as a natural, classical distribution to model count data

• This work introduces and considers the sum of CMP random variables to establish the flexible class of distributions that encompass the Poisson, geometric, Bernoulli, negative binomial, binomial, and CMP distributions as special cases

• Statistical computing for the Poisson and negative binomial distributions are conducted in R (R Core Team 2017) via the function, fitdistr, contained in the MASS package (Venables and Ripley 2002)

## Summary

Introduction

The Poisson distribution is one of the most popular discrete distributions, serving as a natural, classical distribution to model count data. The mean and variance of this random variable are E(Y ) This dispersion index motivates considering the negative binomial distribution as a viable option for addressing data over-dispersion. This work introduces and considers the sum of CMP random variables to establish the flexible class of distributions that encompass the Poisson, geometric, Bernoulli, negative binomial, binomial, and CMP distributions as special cases. The sCMP class further captures the special case distributions of the CMP model: a geometric distribution with success probability, p = 1 − λ, when m = 1, ν = 0, and. The special case where ν = 1 simplifies the sCMP(λ, ν, m) model to the Poisson(mλ) distribution This illustrative example displays Poisson models with respective means equaling 4, 6, and 10. This distinction is key between the CMP distribution and the larger sCMP class – the CMP distribution does not have the invariance property under addition

Moments of the distribution
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