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Construction of k-Variate Survival Functions with Emphasis on the Case k = 3

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Document pages: 20 pages

Abstract: The purpose of this paper is to present a general universal formula for k-variate survival functions for arbitrary k = 2, 3, ..., given all the univariate marginal survivalfunctions. This universal form of k-variate probability distributions was obtained by means of “dependencefunctions” named “joiners” in the text. These joiners determine all theinvolved stochastic dependencies between the underlying random variables. However,in order that the presented formula (the form) represents a legitimate survivalfunction, some necessary and sufficient conditions for the joiners had to befound. Basically, finding those conditions is the main task of this paper. Thistask was successfully performed for the case k = 2 and the main results for the case k = 3 were formulated as Theorem 1 and Theorem 2 in Section 4. Nevertheless,the hypothetical conditions valid for the general k ≥ 4 case were also formulated in Section 3 as the (very convincing)Hypothesis. As for the sufficient conditions for both the k = 3 and k ≥ 4 cases, the full generality was not achieved since two restrictionswere imposed. Firstly, we limited ourselves to the, defined in the text,“continuous cases” (when the corresponding joint density exists and iscontinuous), and secondly we consider positive stochastic dependencies only.Nevertheless, the class of the k-variate distributions which can be constructed is very wide. Thepresented method of construction by means of joiners can be consideredcompetitive to the copula methodology. As it is suggested in the paper thepossibility of building a common theory of both copulae and joiners is quitepossible, and the joiners may play the role of tools within the theory ofcopulae, and vice versa copulae may, for example, be used for finding properjoiners. Another independent feature of the joiners methodology is thepossibility of constructing many new stochastic processes including stationaryand Markovian.

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