Modeling Discrete Bivariate Data with Applications to Failure and Count Data

Abstract

In this study, we propose a new class of flexible bivariate distributions for discrete random variables. The proposed class of distribution is based on the notion of conditional failure rate for a discrete-type random variable. We derive general formulae for the joint distributions belonging to the proposed class that, unlike other discrete bivariate models already proposed in the literature such as the well-known and most popular Holgate's bivariate Poisson distribution, can model both positive and negative dependence. We discuss general statistical properties of the proposed class as well. Specific families of bivariate distributions can be generated from the general class proposed in this paper just by specifying the 'baseline distributions'. Furthermore, specific discrete bivariate distributions belonging to the proposed class are applied to analyze three real data sets, and the results are compared with those obtained from conventional models. in Wiley Online Library i case 1: when component B is at functioning state, the failure rate of component A is the same as the original λ 1 (k) and ii case 2: when component B is at failed state, the failure rate of component A is increased compared with case 1.

Modeling of System Availability and Bayesian Analysis of Bivariate Distribution

“…Mercier and Pham [34] design a bivariate random shock model and review the reliability of the shock model. Lee et al [35] provide a bivariate distribution using conditional failure rate to model discrete random variable. Their method is also flexible to model negative and positive dependence.…”

A bivariate count model with discrete Weibull margins

“…Methods and issues related to the construction of bivariate discrete distributions, which are disseminated in the literature, have been reviewed in [21]; for more recent proposals, see e.g. [16,22]. Whereas the construction of multivariate distributions based on the definition of their joint probability mass or density function poses some difficulties and often results in practical limitations, for example, in the range of possible pairwise correlations; the specification via the marginal distributions and a copula function that provides the dependence structure, is much more straightforward.…”