On a copula product characterizing independence and perfect functional dependence

Abstract

Recently studied dependence measures take values in [0,1], where 0 characterizes independence of X and Y, and 1 characterizes perfect functional (not necessarily monotone) dependence of Y on X. The most prominent example is Chatterjee's rank correlation, which is based on the concept of conditional independence. In contrast, we show that dependence measures such as Wasserstein correlations and rearranged dependence measures are naturally linked to the concept of conditional comonotonicity. This connection is captured by a new copula product T(C)=C\veeΠwhich models conditional comonotonicity and characterizes independence and perfect functional dependence. As a main contribution, we prove that the mapping T acts as a reflection on the class of stochastically increasing copulas. Further, T^2=T \circ T projects a copula onto its increasing rearranged copula. To better understand the behavior of such dependence measures, we also study fixed points, ordering results, and continuity properties of T. Our results demonstrate that conditional comonotonicity is a rather intrinsic feature of dependence measures, whereas conditional independence underlying Chatterjee's rank correlation is a quite exceptional property.

24 pages; 3 figures