Joint distributions of random sets and their relation to copulas

Abstract

Random sets are set-valued random variables. They have been applied in various fields like stochastic geometry, statistics, economics, engineering or computer science, and are often used for modeling uncertainty. In an earlier paper the author has defined joint capacity and joint containment functionals which are multivariate set functions describing the joint distribution of random sets. This paper is concerned with the question how copulas can be used to describe or model the dependence of random sets. It is demonstrated that a joint containment functional can be related to its margins by a family of copulas. Furthermore, the paper provides a first insight how copulas can be used to define joint containment functionals.