Directional footrule-coefficients

TL;DR

This paper introduces a new family of directional dependence measures based on Spearman's footrule within the copula framework, capable of detecting asymmetric dependence patterns in multivariate data.

Contribution

It proposes a novel directional Spearman's footrule coefficient, establishes its theoretical properties, and develops nonparametric estimators for multivariate dependence analysis.

Findings

The coefficients are consistent with classical footrule measures.

They can detect directional dependence patterns.

Explicit expressions demonstrate their effectiveness with known copulas.

Abstract

Rank-based dependence measures such as Spearman's footrule are robust and invariant, but they often fail to capture directional or asymmetric dependence in multivariate settings. This paper introduces a new family of directional Spearman's footrule coefficients for multivariate data, defined within the copula framework to clearly separate marginal behavior from dependence structure. We establish their main theoretical properties, showing full consistency with the classical footrule, including behavior under independence and extreme dependence, as well as symmetry and reflection properties. Nonparametric rank-based estimators are proposed and their asymptotic consistency is discussed. Explicit expressions for several known families of copulas illustrate the ability of the proposed coefficients to detect directional dependence patterns undetected by classical measures.