Premium
Generalized Poisson Distribution: the Property of Mixture of Poisson and Comparison with Negative Binomial Distribution
Author(s) -
Joe Harry,
Zhu Rong
Publication year - 2005
Publication title -
biometrical journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.108
H-Index - 63
eISSN - 1521-4036
pISSN - 0323-3847
DOI - 10.1002/bimj.200410102
Subject(s) - negative binomial distribution , count data , poisson distribution , compound poisson distribution , mathematics , poisson binomial distribution , zero inflated model , quasi likelihood , negative multinomial distribution , binomial distribution , statistics , beta binomial distribution , statistical physics , poisson regression , physics , population , demography , sociology
We prove that the generalized Poisson distribution GP(θ, η) (η ≥ 0) is a mixture of Poisson distributions; this is a new property for a distribution which is the topic of the book by Consul (1989). Because we find that the fits to count data of the generalized Poisson and negative binomial distributions are often similar, to understand their differences, we compare the probability mass functions and skewnesses of the generalized Poisson and negative binomial distributions with the first two moments fixed. They have slight differences in many situations, but their zero‐inflated distributions, with masses at zero, means and variances fixed, can differ more. These probabilistic comparisons are helpful in selecting a better fitting distribution for modelling count data with long right tails. Through a real example of count data with large zero fraction, we illustrate how the generalized Poisson and negative binomial distributions as well as their zero‐inflated distributions can be discriminated. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)