Chapter 3 - General forms of Bivariate survival functions with reliability applications

Abstract

We develop a new general theory for the problem of constructing bivariate probability distributions given two arbitrary marginals. The new approach and the associated theory is competitive to the usually deployed copula methodology. The main area of application of the construction methods, as well as new specific classes of bivariate models, is the reliability of systems with dependent component lifetimes. However, the applicability of our approach goes far beyond that set of reliability problems, especially toward biomedical settings and econometrics. This significantly wide spectrum of possible applications is a result of the high generality of the theory. As it is shown, any arbitrary bivariate survival function can be represented in one, common for all such functions, universal form as the arithmetic product of two marginal survival functions and the defined “dependence function” (the joiner) which impose stochastic dependence. Thus, the main task in any bivariate model's construction is to find a proper joiner given two fixed marginals. For that we formulate sufficient conditions for candidate functions of two variables to be proper joiners “connecting” two, given in advance, marginal survival functions into a bivariate distribution. Based on that criterion we construct three specific, but very wide, classes of bivariate stochastic models which can be applied in reliability as well as in other areas of scientific practice. In Section 3 of this chapter we investigate seven classical bivariate models in light of the newly created theory using new tools. Some interesting new observations on the properties of that seven distributions were obtained independently of the existing knowledge associated with these models.