The W-footrule coefficient: A copula-based measure of countermonotonicity

Abstract

We introduce the $W$-footrule coefficient ${\Phi }_C$, a copula-based coefficient of negative association defined as the $L^1$-distance to the countermonotonic copula $W$. We prove that Gini's gamma admits the decomposition ${\gamma }_C=\frac{2}{3}({\phi }_C+{\Phi }_C)$, linking it to Spearman's footrule ${\phi }_C$. A rank-based estimator is introduced, with its strong consistency and asymptotic normality established via the functional delta method. Monte Carlo simulations confirm the estimator's finite-sample validity and its sensitivity to negative dependence structures.