A Useful Distribution for Fitting Discrete Data: Revival of the COM-Poisson
A useful discrete distribution (the Conway-Maxwell-Poisson, or COM-Poisson) is revived and its statistical and probabilistic properties are introduced and explored. The COMPoisson is a two-parameter extension of the Poisson distribution that generalizes some wellknown discrete distributions (Poisson, Bernoulli, Geometric). It also leads to the generalization of distributions derived from these discrete distributions (viz. the Binomial and Negative Binomial). We describe three methods for estimating the COM-Poisson’s parameters: The first is a fast simple weighted least squares method, which leads to estimates that are sufficiently accurate for practical purposes. The second method, using maximum likelihood, can be used to refine the initial estimates. This method requires iterations and is more computationally intensive. The third estimation method is Bayesian. Using the conjugate prior, the posterior density of the COM-Poisson parameters is easily computed. The COM-Poisson is a flexible distribution that can account for over/under dispersion commonly encountered in count data. We also explore two sets of real world data demonstrating the flexibility and elegance of the COM-Poisson in fitting count data which do not seem to follow the Poisson distribution.