On a copula product linking Wasserstein correlations and rearranged dependence measures

TL;DR

This paper explores the connection between Wasserstein correlations and rearranged dependence measures through a copula product, revealing intrinsic features of dependence and independence in statistical measures.

Contribution

It introduces a copula product linking two classes of dependence measures and analyzes its properties, providing new insights into their structural relationships.

Findings

T acts as a reflection on stochastically increasing copulas.

T^2 projects a copula onto its increasing rearranged form.

Conditional comonotonicity is intrinsic to dependence measures.

Abstract

Recent research in statistics has focused on dependence measures kappa(Y,X) taking values in [0, 1], where 0 characterizes independence of X and Y, and 1 perfect functional dependence of Y on X. One class of such measures consists of the optimal transport-based Wasserstein correlations introduced by Wiesel. Another class comprises the rearranged dependence measures studied by Strothmann, Dette, and Siburg. While the constructions of Wasserstein correlations and rearranged dependence measures seem to be fundamentally different, we show that they are connected by a copula product T (C) = C v {\Pi} that models conditional comonotonicity. As a main contribution, we prove that the mapping T acts as a reflection on the class of stochastically increasing copulas, whereas T^2 = T \circ T projects a copula onto its increasing rearranged copula. We further study fixed points, ordering results, and continuity properties of T to better understand the interplay between these classes of dependence measures. Our results demonstrate that conditional comonotonicity is an intrinsic feature of dependence measures, whereas conditional independence underlying Chatterjee's rank correlation is a rather exceptional property.