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 survival functions. This universal form of k-variate probability distributions was obtained by means of “dependence functions” named “joiners” in the text. These joiners determine all the involved stochastic dependencies between the underlying random variables. However, in order that the presented formula (the form) represents a legitimate survival function, some necessary and sufficient conditions for the joiners had to be found. Basically, finding those conditions is the main task of this paper. This task 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 restrictions were imposed. Firstly, we limited ourselves to the, defined in the text, “continuous cases” (when the corresponding joint density exists and is continuous), and secondly we consider positive stochastic dependencies only. Nevertheless, the class of the k-variate distributions which can be constructed is very wide. The presented method of construction by means of joiners can be considered competitive to the copula methodology. As it is suggested in the paper the possibility of building a common theory of both copulae and joiners is quite possible, and the joiners may play the role of tools within the theory of copulae, and vice versa copulae may, for example, be used for finding proper joiners. Another independent feature of the joiners methodology is the possibility of constructing many new stochastic processes including stationary and Markovian.
Highlights
This work includes a continuation of our previous papers [1] [2] [3] and [4] on analysis and construction methods of multivariate survival functions, given their univariate marginals.This subject was widely developed in the literature since the late 1950s and 60s until nowadays
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 survival functions
The hypothetical conditions valid for the general k ≥ 4 case were formulated in Section 3 as the Hypothesis
Summary
This work includes a continuation of our previous papers [1] [2] [3] and [4] on analysis and construction methods of multivariate survival functions, given their univariate marginals. As we point out, there is a possibility to formulate a common general theory of both copula and joiner representations of bivariate and, possibly, k-variate (k ≥ 3) probability distributions. 3 and 4 since for the k-variate (k ≥ 3) survival functions we restrict our attention to “bi-dependence” only, which means only bivariate joiners may be different than 1 (see, [2]) This restriction dramatically simplifies theory of k-variate distributions as compared with the general theory (for arbitrary k) developed in [2]. We defined a class of discrete time (k represents “time”) stochastic processes with a variety of interesting special cases Such processes need not to be very complicated if we assume that most of the bivariate joiners reduce to 1. Extension to models comprising negative dependencies is quite possible, but requires a more complex theory
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