On a General Class of Discrete Bivariate Distributions
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On a General Class of Discrete Bivariate Distributions Debasis Kundu Indian Institute of Technology Kanpur, Kanpur, India Abstract In this paper we develop a general class of bivariate discrete distributions. The basic idea is quite simple. The marginals are obtained by taking the random geometric sum of the baseline random variables. The proposed class of distributions is a flexible class of bivariate discrete distributions in the sense the marginals can take variety of shapes. The probability mass functions of the marginals can be heavy tailed, unimodal as well as multimodal. It can be both over dispersed as well as under dispersed. We discuss different properties of the proposed class of bivariate distributions. The proposed distribution has some interesting physical interpretations also. Further, we consider two specific base line distributions: Poisson and negative binomial distributions for illustrative purposes. Both of them are infinitely divisible. The maximum likelihood estimators of the unknown parameters cannot be obtained in closed form. They can be obtained by solving three and five dimensional non-linear optimizations problems, respectively. To avoid that we propose to use expectation maximization algorithm, and it is observed that the proposed algorithm can be implemented quite easily in practice. We have performed some simulation experiments to see how the proposed EM algorithm performs, and it works quite well in both the cases. The analysis of one real data set has been performed to show the effectiveness of the proposed class of models. Finally, we discuss some open problems and conclude the paper. AMS (2000) subject classification. Primary 62F10; Secondary 62F03, 62H12. Keywords and phrases. Discrete distributions, Joint probability mass function, Bivariate generating function, Infinite divisibility, Method of moment estimators
1 Introduction An extensive amount of work has been done introducing different bivariate discrete distributions, analyzing their properties and developing different estimation procedures. Special attention has been paid on bivariate geometric distributions and bivariate Poisson distributions, see for example Kocherlakota and Kocherlakota (1992), Kocherlakota (1995), Basu and
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D. Kundu
Dhar (1995), Kumar (2008), Kemp (2013), Lee and Cha (2014), Nekoukhou and Kundu (2017) and Kundu and Nekoukhou (2018) and see the references cited therein. Recently, Lee and Cha (2015) proposed two classes of discrete bivariate distributions and discussed their properties. Their idea is based on the minimum and maximum of two independent non-identically distributed random variables. The idea is quite simple, and it produces different unimodal shapes of bivariate discrete distributions. Unfortunately, because of the nonidentical distributions, the joint probability mass functions (PMFs) or the marginal PMFs may not be in a convenient form. It makes it difficult to compute the estimates of the unknown parameters, and to derive different properties. Moreover, the marginals produced by the method of Le
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