On Minimal Copulas under the Concordance Order

TL;DR

This paper explores the concept of minimal copulas under the concordance order, identifying conditions for minimality and demonstrating their importance in optimization problems involving dependence measures.

Contribution

It introduces new sufficient and necessary conditions for minimal copulas, highlighting their role in dependence modeling and optimization.

Findings

Minimal copulas are characterized by d-countermonotonicity.

Every continuous, concordance order preserving functional is minimized by some minimal copula.

Minima of Spearman's rho coincide with those of Kendall's tau.

Abstract

In the present paper, we study extreme negative dependence focussing on the concordance order for copulas. With the absence of a least element for dimensions $d\ge$ 3, the set of all minimal elements in the collection of all copulas turns out to be a natural and quite important extreme negative dependence concept. We investigate several sufficient conditions and we provide a necessary condition for a copula to be minimal: The sufficient conditions are related to the extreme negative dependence concept of d-countermonotonicity and the necessary condition is related to the collection of all copulas minimizing multivariate Kendall's tau. The concept of minimal copulas has already been proved to be useful in various continuous and concordance order preserving optimization problems including variance minimization and the detection of lower bounds for certain measures of concordance. We substantiate this key role of minimal copulas by showing that every continuous and concordance order preserving functional on copulas is minimized by some minimal copula and, in the case the continuous functional is even strictly concordance order preserving, it is minimized by minimal copulas only. Applying the above results, we may conclude that every minimizer of Spearman's rho is also a minimizer of Kendall's tau.