Properties and inferential issues of a bivariate version of the geometric distribution

In recent years, the construction of bivariate (and multivariate) discrete distributions has attracted much interest, since stochastic models for correlated count data find application in many fields. Several authors have discussed the problem of constructing a bivariate version of a given univariate distribution, although there is no universally accepted criterion for producing a unique distribution which can unequivocally be called the bivariate analogue of a univariate distribution. In this paper, we revise a bivariate geometric model, introduced by Roy [1993], which is characterized by locally constant bivariate failure rates. We highlight its close relationship with Gumbel’s bivariate exponential distribution [Gumbel 1960] and then we focus on four aspects of this model that have not been investigated so far: 1) Pearson’s correlation and its range, 2) conditional distributions and pseudo-random simulation, 3) parameter estimation, and 4) stress-strength reliability parameter. A Monte Carlo simulation study is carried out in order to assess the performance of the different estimators proposed; an application to real data, along with a comparison with alternative bivariate discrete models, is provided as well.