On Minimal Copulas under the Concordance Order

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 ≥ 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.

Dissimilarity functions for rank-invariant hierarchical clustering of continuous variables

“…Remark 3.1. Notice that the single, average and complete extended dissimilarity functions induced by d 1,1 := f • ρ, where ρ is the pairwise Spearman's correlation and f is a strictly decreasing function, are strictly monotonically decreasing with respect to the lower orthant order (see, e.g., [2]).…”

Living on the Edge: An Unified Approach to Antithetic Sampling

“…Multivariate Kendall' τ and Spearman's ρ are multivariate measures of concordance introduced in Joe (1990) as a generalization of the well known bivariate measures. Fuchs et al (2018); Ahn and Fuchs (2020) show that d-CTM vectors have the same minimal multivariate Kendall's τ but different Spearman's ρ.…”

“…We compare different strict d-CTM constructions using the concordance order. According to Ahn and Fuchs (2020), all strict d-CTM have minimal multivariate Kendall's τ , but they can have different multivariate Spearman's ρ values. For example, Gaffke and Rüschendorf (1981), and Arvidsen and Johnsson (1982) proposals have the same values for Kendall's τ , but different ones for Spearman's ρ.…”

A few generalizations of Kendall’s tau. Part I: Construction

“…Theoretical studies on concordance measures, both bivariate and multivariate, have focused on their construction and properties (see, e.g., [3,10,12,15,16,23,33]), on the precise bounds for the values of such measures, in particular when some additional information is known, or for other objects given the value of some concordance measure (see, e.g., [1,7,20,21,36]).…”