On bivariate copula-based geometric models with application to reliability

In many scientific fields, researchers are concerned with multivariate random variables. Although quantities measured on a continuous scale are more frequent, nevertheless multivariate count data often arise in several contexts (statistical process control, epidemiology, failure and reliability analysis, etc). Such data are frequently modelled through the multivariate Poisson distribution, based on a general multivariate reduction scheme, which however suffers from some practical limits. Various methods have been proposed for constructing new alternative multivariate discrete random variables that can be used as viable alternatives. One of the most straightforward is that based on joining arbitrary univariate discrete distributions through a copula function; this method allows for both flexible dependence structure and flexible choice of margins. In this work, we discuss some bivariate geometric distributions derived according to different copula functions, examining their properties, with particular regard to reliability concepts, and fitting them to real datasets taken from the literature.