An Exceptional Generalization of the Poisson Distribution

Per-Erik Hagmark

Department of Mechanics and Design, Tampere University of Technology, Tampere, Finland.

DOI: 10.4236/ojs.2012.23039                

Abstract

A new two-parameter count distribution is derived starting with probabilistic arguments around the gamma function and the digamma function. This model is a generalization of the Poisson model with a noteworthy assortment of qualities. For example, the mean is the main model parameter; any possible non-trivial variance or zero probability can be attained by changing the other model parameter; and all distributions are visually natural-shaped. Thus, exact modeling to any degree of over/under-dispersion or zero-inflation/deflation is possible.

Keywords

Count Data; Gamma Function; Poisson Generalization; Discretization; Modeling; Over/Under-Dispersion; Zero-Inflation/Deflation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. Castillo and M. Perez-Casany, “Over-Dispersed and Under-Dispersed Poisson Generalizations,” Journal of Statistical Planning and Inference, Vol. 134, No. 2, 2005, pp. 486-500. doi:10.1016/j.jspi.2004.04.019
[2]	P. C. Consul and G. C. Jain, “A Generalization of the Poisson Distribution,” Technometrics, Vol. 15, No. 4, 1973, pp. 791-799. doi:10.2307/1267389
[3]	N. L. Johnson, S. Kotz and A. W. Kemp, “Univariate Discrete Distributions,” 2nd Edition, John Wiley & Sons, New York, 1992.
[4]	R. W. Conway and W. L. Maxwell, “A Queuing Model with State Dependent Service Rates,” Journal of Industrial Engineering, Vol. 12, 1962, pp. 132-136.
[5]	G. Morlat, “Sur Une Généralisation de la loi de Poisson,” Comptes Redus, Vol. 235, 1952, pp. 933-935.
[6]	P.-E. Hagmark, “On Construction and Simulation of Count Data Models,” Mathematics and Computers in Simulation, Vol. 77, No. 1, 2008, pp. 72-80. doi:10.1016/j.matcom.2007.01.037
[7]	L. Gordon, “A Stochastic Approach to the Gamma Function,” The American Mathematical Monthly, Vol. 101, No. 9, 1994, pp. 858-865.
[8]	A. J. Walker, “An Efficient Method for Generating Discrete Random Variables with General Distributions,” ACM Transactions on Mathematical Software, Vol. 3, 1977, pp. 253-256. doi:10.1145/355744.355749