Sklar's theorem for minitive belief functions

Abstract

In a former paper, the author has investigated how copulas can be used to express the dependence relation between two random sets. It has been proven that a joint belief function is related to its marginal belief functions by a family of copulas and that, in general, a single copula is not sufficient. In this paper the results are investigated under the assumption that the involved belief functions are minitive which corresponds to the important case where the associated random sets are consonant. It is proven that under this additional assumption a single copula is sufficient to express the dependence relation. In other words, this means that Sklar's theorem remains valid if joint and marginal distribution functions are replaced by joint and marginal minitive belief functions.

Joint distributions of random sets and their relation to copulas

“…This could comprise particular conditions on copulas and/or containment functionals. In [17] a first result in this direction is presented. The paper contains a version of Sklar's theorem for minitive containment functionals.…”

“…The latter correspond to lower probabilities of consonant random sets. A detailed discussion is given in [17] where a version of Sklar's theorem for minitive containment functionals is proven.…”

Complementary Lipschitz continuity results for the distribution of intersections or unions of independent random sets in finite discrete spaces

“…The difficulty in this regard is that one can no longer just work with marginal inclusion or containment functionals but with multivariate ones [36] that do not factorize as elementwise products of marginal functionals. Recent work [37,38,39] generalizing Sklar's theorem on copulas to joint or multivariate capacity functionals of random sets may be useful in this quest because it gives an explicit connection between marginal functionals and multivariate ones. However, in contrast to point-valued random variables, a family of copulas is necessary to characterize this link.…”

“…Similarly, K can be retrieved as the L ∞ norm based distances between the contour functions of m 1 and m 2 . This observation raises the following question: can we build other relevant degrees of conflict in the same fashion as in (38) but using other distances than d pl,∞ ? We try to provide some answers to this question in the next paragraphs when the examined distances are consistent with ∩ .…”

Constructing copulas from shock models with imprecise distributions

“…As a matter of fact, it turns out that sets of copulas are needed to describe such dependence instead of a single copula. Moreover, he also shows in [56] that in the special case of minitive belief functions Sklar's theorem actually holds, which means that a single copula is sufficient to express the dependence relation.…”