# Generalized Poisson Distribution: the Property of Mixture of Poisson and Comparison with Negative Binomial Distribution

Generalized Poisson Distribution: the Property of Mixture of Poisson and Comparison with Negative... 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) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Biometrical Journal Wiley

# Generalized Poisson Distribution: the Property of Mixture of Poisson and Comparison with Negative Binomial Distribution

, Volume 47 (2) – Apr 1, 2005
11 pages

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# References (17)

Publisher
Wiley
ISSN
0323-3847
eISSN
1521-4036
DOI
10.1002/bimj.200410102
Publisher site
See Article on Publisher Site

### Abstract

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)

### Journal

Biometrical JournalWiley

Published: Apr 1, 2005