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A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution
Author(s) -
Shmueli Galit,
Minka Thomas P.,
Kadane Joseph B.,
Borle Sharad,
Boatwright Peter
Publication year - 2005
Publication title -
journal of the royal statistical society: series c (applied statistics)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.205
H-Index - 72
eISSN - 1467-9876
pISSN - 0035-9254
DOI - 10.1111/j.1467-9876.2005.00474.x
Subject(s) - poisson distribution , mathematics , negative binomial distribution , overdispersion , compound poisson distribution , beta binomial distribution , count data , poisson binomial distribution , binomial distribution , quasi likelihood , univariate distribution , zero inflated model , probability distribution , statistical physics , statistics , poisson regression , physics , population , demography , sociology
Summary. A useful discrete distribution (the Conway–Maxwell–Poisson distribution) is revived and its statistical and probabilistic properties are introduced and explored. This distribution is a two‐parameter extension of the Poisson distribution that generalizes some well‐known discrete distributions (Poisson, Bernoulli and geometric). It also leads to the generalization of distributions derived from these discrete distributions (i.e. the binomial and negative binomial distributions). We describe three methods for estimating the parameters of the Conway–Maxwell–Poisson distribution. 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 parameters of the Conway–Maxwell–Poisson distribution is easily computed. It is a flexible distribution that can account for overdispersion or underdispersion that is commonly encountered in count data. We also explore two sets of real world data demonstrating the flexibility and elegance of the Conway–Maxwell–Poisson distribution in fitting count data which do not seem to follow the Poisson distribution.