The exact region determined by Spearman's footrule, Gini's gamma and Kendall's tau

TL;DR

This paper characterizes the exact region defined by three concordance measures—Spearman's footrule, Gini's gamma, and Kendall's tau—using copula-based methods, enhancing understanding of their interrelations.

Contribution

It provides the precise description of the region determined by these measures and formulas for related concordance measures and Chatterjee's xi.

Findings

Describes the exact region for three concordance measures.

Provides formulas for five main concordance measures.

Includes analysis of ordinal sums of copulas.

Abstract

Concordance measures are used to express the degree of association between random variables. Practitioners may use several distinct concordance measures to narrow the space of possible dependence structures. Consequently, the relations between different (weak) concordance measures have been extensively studied in recent years. The goal of this paper is to study the relation between Kendall's tau, Gini's gamma and Spearman's footrule. In particular, we describe the exact region determined by these three measures, using shuffles of $M$ and ordinal sums of copulas. We also provide the formulas for five main (weak) concordance measures and Chatterjee's xi of ordinal sums of copulas.