Extremal Dependence Concepts

TL;DR

This paper reviews extremal dependence concepts, introduces a new notion of negative dependence, and discusses their roles in probabilistic modeling and optimization, highlighting the importance of negative dependence alongside positive dependence.

Contribution

It provides a comprehensive review of extremal dependence concepts, introduces a novel notion of negative dependence, and links these ideas to probabilistic optimization problems.

Findings

Extremal positive dependence is well-defined for any dimension.

Various notions of extremal negative dependence exist and are reviewed.

A new general concept of negative dependence is proposed.

Abstract

The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges, and recent developments in extremal dependence concepts have drawn a lot of attention to probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative dependence given in literature and introduce a novel notion, which in a general sense includes the existing ones as particular cases. Even if much of the literature on dependence is focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements. This is not only for historical reasons. Rearrangement functions describe the relationship between random variables in a completely deterministic way, allow a deeper understanding of dependence itself, and have several advantages on the approximation of solutions in a broad class of optimization problems.