On Bivariate Discrete Weibull Distribution

TL;DR

This paper introduces a new bivariate discrete Weibull distribution, explores its properties, and demonstrates its effectiveness through data analysis using maximum likelihood and Bayesian methods.

Contribution

It is the first to propose a bivariate discrete Weibull distribution, providing properties, estimation algorithms, and practical data analysis applications.

Findings

Distribution is highly flexible with four parameters.

The nested EM algorithm efficiently estimates parameters.

Bayesian estimates are effectively computed with Gibbs sampling.

Abstract

Recently, Lee and Cha (2015, `On two generalized classes of discrete bivariate distributions', {\it American Statistician}, 221 - 230) proposed two general classes of discrete bivariate distributions. They have discussed some general properties and some specific cases of their proposed distributions. In this paper we have considered one model, namely bivariate discrete Weibull distribution, which has not been considered in the literature yet. The proposed bivariate discrete Weibull distribution is a discrete analogue of the Marshall-Olkin bivariate Weibull distribution. We study various properties of the proposed distribution and discuss its interesting physical interpretations. The proposed model has four parameters, and because of that it is a very flexible distribution. The maximum likelihood estimators of the parameters cannot be obtained in closed forms, and we have proposed a very efficient nested EM algorithm which works quite well for discrete data. We have also proposed augmented Gibbs sampling procedure to compute Bayes estimates of the unknown parameters based on a very flexible set of priors. Two data sets have been analyzed to show how the proposed model and the method work in practice. We will see that the performances are quite satisfactory. Finally, we conclude the paper.