A generalized Conway–Maxwell–Poisson distribution which includes the negative binomial distribution

Abstract

The Conway–Maxwell–Poisson (COM-Poisson) distribution with two parameters was originally developed as a solution to handling queueing systems with state-dependent arrival or service rates. This distribution generalizes the Poisson distribution by adding a parameter to model over-dispersion and under-dispersion and includes the geometric distribution as a special case and the Bernoulli distribution as a limiting case. In this paper, we propose a generalized COM-Poisson (GCOM-Poisson) distribution with three parameters, which includes the negative binomial distribution as a special case, and can become a longer-tailed model than the COM-Poisson distribution. The new parameter plays the role of controlling length of tail. The GCOM-Poisson distribution can become a bimodal distribution where one of the modes is at zero and is applicable to count data with excess zeros. Estimation methods are also discussed for the GCOM-Poisson distribution.

Introduction

In statistical research, it is important to select an adequate distribution to describe the observed variation of counts. The Poisson distribution is a classically utilized model for analyzing count data. However, it has a serious restriction that the variance is equal to the mean or, equivalently, the index of dispersion (the ratio of variance to mean) is one, because observed count data do not satisfy the equality of the sample mean and variance in many cases. For many observed count data, it is common to have the sample variance to be greater or smaller than the sample mean which are referred to as over-dispersion and under-dispersion, respectively, relative to the Poisson distribution. Information on dispersion is useful for selecting an appropriate model to count data. For example, the negative binomial distribution is often selected for over-dispersed data and the binomial distribution is for under-dispersed data.

Shmueli et al. [7] have revived the Conway–Maxwell–Poisson (COM-Poisson for short) distribution, originally developed by Conway and Maxwell [2] as a solution to handling queueing systems with state-dependent arrival or service rates, and indicated its flexibility to adapt to over- and under-dispersions. The COM-Poisson distribution has the probability mass function (pmf)

$$P(x)=\frac{{\lambda }^x}{{(x!)}^r}\frac{1}{Z(\lambda \text{,}r)}\text{,}\text{where}Z(\lambda \text{,}r)={\displaystyle \sum _{k=0}^{\infty }}\frac{{\lambda }^k}{{(k!)}^r}$$

for $r>0$ and $\lambda >0$ and reduces to the geometric distribution when $r\to 0$ and $0<\lambda <1$ and the Bernoulli distribution when $r\to \infty$. This means that the COM-Poisson distribution can become an over- or under-dispersed model. This flexibility greatly expands the types of problems for which the COM-Poisson distribution can be used to model count data.

In empirical modeling, the length of the tail parts of the distribution is an important factor. The negative binomial distribution is a generalized form of the geometric distribution and becomes a longer-tailed distribution. This paper proposes a generalization of the COM-Poisson (GCOM-Poisson for short) distribution, which includes the negative binomial distribution as a special case and, therefore, can become a longer-tailed model than the original COM-Poisson distribution. Moreover, the GCOM-Poisson can become a bimodal distribution where one of the modes is at zero and, therefore, can be adapted to count data with excess zeros. The flexibility of the dispersion and the length of the tail and applicability to excess zeros make the proposed distribution more versatile than the COM-Poisson distribution.

This paper is arranged as follows. The definition of the GCOM-Poisson distribution with some properties is given in Section 2. In Section 3, we consider methods of estimation for fitting the proposed distribution to real data sets and numerical examples using the methods are given in Section 4. Finally, our conclusion is given in Section 5.