Theoretical and Reliability Aspects of Multivariate Probability Distributions in their Universal Form (original) (raw)

2019

Abstract

We consider a comprehensive solution to the problem of finding the joint k-variate probability distributions of random vectors (X1, … ,Xk), given all the univariate marginals for an arbitrary k ≥ 3. The one general and universal analytic form of all the solutions, given the fixed univariate marginals, was given in the proven theorem. In order to choose among all its particular realizations one needs to determine proper "dependence functions" (joiners) which 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 [5], [6]. In applications, such as system reliability modeling and other, among all the available k-variate solutions one needs to chose those that may fit to a particular data and, after that, testify the chosen models by proper statistical methods. The theoretical aspect of the main model, as given by formula (3) in section 2, mainly relies on the existence of one [for the given 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 an 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 is provided.

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