A General (Universal) Form of Multivariate Survival Functions in Theoretical and Modeling Aspect of Multicomponent System Reliability Analysis

Abstract

One of important classical reliability problems can be formulated as follows. Given a k-component system (k = 2, 3, …) with series reliability structure where the life-times X₁, …, X_k of the components are nonnegative stochastically dependent random variables. In order to determine the system’s (as a whole) reliability function as well as for some system maintenance analysis, one needs to find or construct a proper joint probability distribution of the random vector (X₁, …,X_k) which may be expressed in terms of the joint reliability (survival) function. Numerous particular solutions for this problem are present in the literature, to mention only [Freund in J Am Stat Assoc 56:971–77, 1961 1, Gumbel in J Am Stat Assoc 55:698–707, 1960 2, Marshall and Olkin in J Appl Probab 4:291–303, 1967 3]. Many other, not directly associated with reliability, k-variate probability distributions were invented [Kotz et al. in Continuous Multivariate Distributions, Wiley, New York, 2000 4]. Some of them later turned out to be applicable to the above considered reliability problem. However, the need for proper models still highly exceeds the existing supply. In this chapter we present not only particular bivariate and k-variate new models, but, first of all, a general method for their construction competitive to the copula methodology [Sklar A in Fonctions de repartition a n dimensions et leurs marges, Publications de l’Institut de Statistique de l’Universite de Paris, pp. 229–231, 1959 5]. The method follows the invented universal representation of any bivariate and k-variate survival function different from the corresponding copula representation. A comparison of our representation with the one given by copulas is provided. Also, some new bivariate models for 2-component series systems are presented. Possible applications of our models and methods may go far beyond the reliability context, especially toward bio-medical and econometric areas.