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

TL;DR

The paper establishes the exact region of feasible triples $(\phi, \gamma, \tau)$ for copulas using Spearman's footrule, Gini's gamma, and Kendall's tau by developing concordance formulas for ordinal sums and proving that boundary points are achieved by shuffles of $M$. It defines a convex polyhedron $\Omega$ with seven faces that captures all attainable triplets, and demonstrates constructive copula examples attaining boundary points. The analysis connects known two-measure regions via projections and quantifies the region's volume, showing the rank between $\tau$ and the pair $(\phi,\gamma)$ is stronger than with $\beta$. The results extend the study of multi-measure concordance relations and provide explicit tools for constructing copulas with prescribed triplet values.

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.

Figures (5)

• Figure 1: The polyhedron $\Omega$ defined in \ref{['eq:Omega']}. Theorem \ref{['thm:main']} shows that $\Omega$ is precisely the region determined by $\phi$, $\gamma$ and $\tau$.
• Figure 2: The mass distribution of copula $C_b$ (left) and the graphs of functions $\delta_{C_b}$ and $\omega_{C_b}$ (right) from the proof of Lemma \ref{['lem:F6']}.
• Figure 3: The mass distribution of copula $D_b$ (left) and the graphs of functions $\delta_{D_b}$ and $\omega_{D_b}$ (right) from the proof of Lemma \ref{['lem:F4']}.
• Figure 4: The mass distribution of copula $G_b$ (left) and the graphs of functions $\delta_{G_b}$ and $\omega_{G_b}$ (right) from Example \ref{['ex:F7']}.
• Figure 5: The mass distribution of copula $L_{a,b}$ (left) and the graphs of functions $\delta_{L_{a,b}}$ and $\omega_{L_{a,b}}$ (right) from Example \ref{['ex:F5']}.

Theorems & Definitions (32)

• Proposition 2.1: KoBuMoKoBuSt
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Lemma 3.4
• proof
• Proposition 3.5