2 - The Conway–Maxwell–Poisson (COM–Poisson) Distribution

Summary

This chapter defines the COM–Poisson distribution in greater detail, discussing its associated attributes and computing tools available for analysis. This chapter first details how the COM–Poisson distribution was derived, and then describes the probability distribution, and introduces computing functions available in R that can be used to determine various probabilistic quantities of interest, including the normalizing constant, probability and cumulative distribution functions, random number generation, mean, and variance. The chapter then outlines the distributional and statistical properties associated with this model, and discusses parameter estimation and statistical inference associated with the COM–Poisson model. Various processes for generating random data are then discussed, along with associated available R computing tools. Continued discussion provides reparametrizations of the density function that serve as alternative forms for statistical analyses and model development, considers the COM–Poisson as a weighted Poisson distribution, and details discussion describing the various ways to approximate the COM–Poisson normalizing function.