A New Weighted Poisson Distribution For Over-and Under- Dispersion Situations.

In this paper, we propose a four-parameter weighted Poisson distribution that includes and generalizes the weighted Poisson distribution proposed by Castillo and Pérez-Casany and the Conway-Maxwell-Poisson distribution, as well as other well-known distributions. It is a distribution that is a member of the exponential family and is an exponential combination formulation between the weighted Poisson distribution proposed by Castillo and Pérez-Casany and the Conway-Maxwell-Poisson distribution. This new distribution with an additional parameter of dispersion is more flexible, and the Fisher dispersion index can be greater than, equal to, or less than one. This last property allows it to model over-dispersed data as well as under-dispersed or equi-dispersed data. Many other properties of the new distribution are studied in this article. The parameters are estimated by two methods: the least squares and the maximum likelihood methods. The properties of the estimators are not studied because they follow directly. Two examples of application to real data, taking into account situations of overdispersion and underdispersion, are examined. [ABSTRACT FROM AUTHOR]