Joint distributions of random sets and their relation to copulas

Sklar's theorem for minitive belief functions

“…The essential conclusion from [20] is that a single copula is in general not sufficient to describe the dependence relation between a joint containment functional and its margins. More precisely, for each pair of increasing subfamilies from K c …”

“…One exception where a single copula is sufficient to describe the dependence relation between a joint containment functional and its margins is the case of independence. It turns out that two random sets are independent if and only if for all increasing families their joint containment functional is related to its margins via the product copula [20,Prop. 7], and, conversely, defining a bivariate set function as the product of two univariate containment functionals yields a joint…”

“…In [20] it has been investigated how copulas can be used to characterize the dependence relation between two random sets. The idea was to derive a version of Sklar's theorem where the role of the joint and marginal distribution functions is taken by the joint and marginal containment (or capacity) functionals of the random sets.…”

“…It should be noted that the notion of joint (or multivariate) capacity functionals has been coined in [19] where a characterization of the joint distribution of random sets by multivariate set functions has been presented. It has been proven in [20] that the dependence relation between two random sets can be characterized by a family of copulas and that in contrast to the classical case of random variables a single copula is in general not sufficient. In this paper an additional assumption will be made about the marginal containment functionals: It will be assumed that the latter are minitive.…”

“…In Section 2 basic facts about belief functions, random sets and their joint distribution are reviewed. Section 3 summarizes the most important results from [20] concerning the usage of copulas to characterize the dependence relation between (general) random sets. Section 4 constitutes the main part of the paper.…”