Mathematical Reliability Aspects of Multivariate Probability Distributions

Abstract

This work deals with a comprehensive solution to the problem of finding the joint k-variate probability distributions of random vectors (X1, …,Xk), given all the univariate marginals. The general and universal analytic form of all solutions, given the fixed (but arbitrary) univariate marginals, was given in proven theorem. In order to choose among these solutions, one needs to determine proper “dependence functions” (joiners) that impose specific stochastic dependences among subsets of the set {X1, …,Xk} of the underlying random variables. Some methods of finding such dependence functions, given the fixed marginals, were discussed in our previous papers (Filus and Filus, J Stat Sci Appl 5:56–63, 2017; Filus and Filus, General method for construction of bivariate stochastic processes given two marginal processes. Presentation at 7-th International Conference on Risk Analysis, ICRA 7, Northeastern Illinois University, Chicago, 4 May 2017). In applications, such as system reliability modeling and other, among all the available k-variate solutions, one needs to choose those that may fit particular data, and, after that, test the chosen models by proper statistical methods. The theoretical aspect of the main model, given by formula (7.3) in Sect. 7.2, mainly relies on the existence of one [for any fixed set of univariate marginals] general and universal form which plays the role of paradigm describing the whole class of the k-variate probability distributions for arbitrary k = 2, 3, …. An important fact is that the initial marginals are arbitrary and, in general, each may belong to a different class of probability distributions. Additional analysis and discussion are provided.