Chapter 8 - Multivariate Lifetime Models

Abstract

After having discussed the ageing notions and classes of life distributions based on their properties, the next stage is to propose some basic models that can be used in analysing multivariate failure time data. We concentrate mainly on bivariate distributions whose reliability functions have simple and meaningful mathematical forms. To begin with, we present seven different versions of multivariate geometric distributions. All of them have geometric marginals with the component lives thus possessing no-ageing property, but with different kinds of ageing and dependence properties at the system level. These distributions permit generalizations to more flexible models like the discrete Weibull, gamma, etc., at a later stage. In the Bayesian framework, Schur-constant models are important as they possess the no-ageing property. We study the distributional and reliability aspects in some detail. It is pointed out that in such models, ageing and dependence notions are interrelated. Two forms of bivariate Waring distributions, of which one is Schur-constant and the other possessing linear mean residual life function, are presented subsequently. Various reliability functions and characterizations based on them are studied. A similar work is done relating to negative hypergeometric distribution. This chapter finally concludes with some results on a bivariate Weibull and multivariate Zipf distributions.