Univariate and bivariate geometric discrete generalized exponential distributions

Abstract

In 1997, Marshall and Olkin introduced a very powerful method to introduce an additional parameter to a class of continuous distribution functions that brings more flexibility to the model. They demonstrated their method for the exponential and Weibull classes. In the same paper they briefly indicated its bivariate extension. The main aim of this article is to introduce the same method, for the first time, to the class of discrete generalized exponential distributions both for the univariate and bivariate cases. We investigate several properties of the proposed univariate and bivariate classes. The univariate class has three parameters, whereas the bivariate class has five parameters. It is observed that depending on the parameter values, the univariate class can be zero inflated as well as heavy tailed. We propose to use an expectation–maximization (EM) algorithm to estimate the unknown parameters. Small simulation experiments have been performed to see the effectiveness of the proposed EM algorithm, and a bivariate data set has been analyzed; it is observed that the proposed models and the EM algorithm work quite well in practice.