TITLE:
Construction of k-Variate Survival Functions with Emphasis on the Case k = 3
AUTHORS:
Jerzy K. Filus, Lidia Z. Filus
KEYWORDS:
Construction of Multivariate Probability Distributions via Joiners, Joiner versus Copula Methodology, A Possible Fusion of the Two Construction Methods, k -Variate Survival Function Scheme, k = 3 Case
JOURNAL NAME:
Applied Mathematics,
Vol.11 No.7,
July
28,
2020
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.