A discrete distribution including the Poisson

Abstract

This article presents a new generalization of the Poisson distribution, with the parameters α > 0 and θ > 0, using the Marshall and Olkin (1997 Marshall, A.W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84(3):641–652.Crossref], Web of Science ®] , Google Scholar]) scheme and adding a parameter to the classical Poisson distribution. The particular case of α = 1 gives the Poisson distribution. The new distribution is unimodal and has a failure rate that monotonically increases or decreases depending on the value of the parameter α. After reviewing some of the properties of this distribution, we investigated the question of parameter estimation. Expected frequencies were calculated for two data sets, one with an index of dispersion larger than one and the other with an index of dispersion smaller than one. In both cases the distribution provided a very satisfactory fit.