Random sets, copulas and related sets of probability measures

Abstract

Random sets are random variables whose values are sets. They are frequently interpreted as imprecise observations of random variables. In earlier papers the author has investigated how copulas can be used to describe the dependence between random sets. The aim of this paper is to investigate the probabilistic information induced by two random sets and a copula. Two specific sets of probability measures will be investigated: one consisting of the joint distributions of random variables contained in the random sets all having the same copula, and the other one consisting of all distributions covered by a random set in product space constructed from the marginal random sets and the copula. It is shown that even though, in general, none of the credal sets is a subset of the other one, for the special case when the marginal random sets are probability boxes the former credal set is a subset of the latter and that the lower and upper probabilities coincide on cylindrical sets. In addition, new results on random sets in the real line as well as random intervals and their joint distribution are presented.