Univariate and Bivariate Geometric Discrete Generalized Exponential Distributions

TL;DR

This paper introduces a new class of univariate and bivariate discrete generalized exponential distributions using Marshall and Olkin's method, explores their properties, and demonstrates effective parameter estimation with EM algorithms.

Contribution

It extends Marshall and Olkin's method to discrete generalized exponential distributions, providing new models with practical estimation techniques.

Findings

The univariate class can be zero inflated or heavy tailed.

The EM algorithm effectively estimates parameters.

Models perform well on real data.

Abstract

Marshall and Olkin (1997, Biometrika, 84, 641 - 652) introduced a very powerful method to introduce an additional parameter to a class of continuous distribution functions and hence it brings more flexibility to the model. They have demonstrated their method for the exponential and Weibull classes. In the same paper they have briefly indicated regarding its bivariate extension. The main aim of this paper 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 both zero inflated as well as heavy tailed. We propose to use 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 and it is observed that the proposed models and the EM algorithm work quite well in practice.