A New General Class of Discrete Bivariate Distributions Constructed by the Usual Stochastic Order

Abstract

In this article, we develop a new general class of discrete bivariate distributions that can model the effect of the so-called "load-sharing configuration." Under such load-sharing configuration, after the failure of one component, the surviving component has to shoulder extra load, which eventually results in its failure at an earlier time than what is expected under the case of independence. To model such effect, in this article, the residual lifetime of the surviving component is assumed to be shortened according to the usual stochastic order. We derive the joint probability mass function, the joint survival function and the marginal distributions. The identifiability of the proposed model is thoroughly investigated. We discuss a bivariate ageing property of the developed class. It will be seen that the obtained joint distribution can be expressed in terms of existing underlying distributions, which increases the applicability of the developed bivariate distributions. It will also be shown that the developed class has a high degree of flexibility in the sense that numerous families of distributions can be generated just by specifying different underlying distributions and different parameter functions for modeling stochastic dependence. Some specific families of discrete bivariate distributions which can be usefully applied in practice are obtained, and their usefulness is illustrated by some real dataset analyses.