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The exact region between Chatterjee's and Blest's rank correlations

Marcus Rockel

Abstract

Exact regions between rank correlations describe the set of all pairs of values that two dependence measures can attain simultaneously on the same copula and thus yield sharp inequalities between them. In this paper, we determine the exact region between Chatterjee's rank correlation $ξ$ and Blest's rank correlation $ν$ over the class of all bivariate copulas. Our approach is based on a constrained optimization problem whose solution is characterized by Karush--Kuhn--Tucker conditions. This leads to a novel extremal copula family that uniquely traces the boundary of the region. For this family, we derive closed-form expressions for both $ξ$ and $ν$, which provide an explicit parametrization of the exact attainable region.

The exact region between Chatterjee's and Blest's rank correlations

Abstract

Exact regions between rank correlations describe the set of all pairs of values that two dependence measures can attain simultaneously on the same copula and thus yield sharp inequalities between them. In this paper, we determine the exact region between Chatterjee's rank correlation and Blest's rank correlation over the class of all bivariate copulas. Our approach is based on a constrained optimization problem whose solution is characterized by Karush--Kuhn--Tucker conditions. This leads to a novel extremal copula family that uniquely traces the boundary of the region. For this family, we derive closed-form expressions for both and , which provide an explicit parametrization of the exact attainable region.
Paper Structure (7 sections, 12 theorems, 128 equations, 2 figures, 1 table)

This paper contains 7 sections, 12 theorems, 128 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. A novel copula family for the boundary of the (xi,nu)-region
  3. Maximizing nu at fixed xi
  4. Proofs
  5. Proofs for the copula construction and Theorem \ref{['thm:closed']}
  6. Proof of Theorem \ref{['thm:KKT']}
  7. Proof of Theorem \ref{['thm:region']}

Key Result

Theorem 1.1

Let $b>0$ and define $\Xi,\mathrm{N}:(0,\infty)\to\mathbb{R}$ by with $\gamma:=\sqrt{\frac{b-1}{b}}$ for $b>1$. Then the exact $(\xi,\nu)$-region is convex, closed, and satisfies where we use the endpoint conventions $\Xi(0)=\mathrm{N}(0)=0$ and $\Xi(\infty)=\mathrm{N}(\infty)=1$. Furthermore, the upper and lower curved boundary branches of $\mathcal{R}_{\xi,\nu}$ are traced uniquely by the copu

Figures (2)

  • Figure 1: The convex and closed attainable $(\xi,\nu)$-region. The boundary curve is uniquely traced by the copula family $(C^{\xi,\nu}_b)_{b\in\mathbb{R}\setminus\{0\}}$ introduced in Section \ref{['sec:novel-copula']}. $\Pi(u,v):=uv$, $M(u,v):=\min\{u,v\}$ and $W(u,v):=\max\{u+v-1, 0\}$, for $u,v\in[0,1]$, denote the independence copula and the upper and lower Fréchet--Hoeffding copulas, respectively.
  • Figure 2: Conditional distribution plots for the copula $C_b$ defined in \ref{['eq:clamped']} with $b=0.5$ (left), $b=1$ (middle), and $b=5$ (right). A large value of Blest's rank correlation for a fixed value of Chatterjee's rank correlation is achieved by allocating the functional dependence more heavily towards the top ranks (left side of each plot).

Theorems & Definitions (21)

  • Theorem 1.1: Exact $(\xi,\nu)$-region in the $b$--parametrization
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3: Closed-form $\xi$ and $\nu$ for $C_b$
  • Lemma 3.1: KKT framework, bonnans2013perturbationrockel2025exact
  • Lemma 3.2: A characterization of copulas, ansari2025exact
  • Theorem 3.4: Solution to Optimization Problem \ref{['opt:main']}
  • proof : Proof of Lemma \ref{['lem:unique_q']}
  • proof : Proof of Lemma \ref{['lem:cb_copula']}
  • Lemma 4.1: Sections and one–dimensional forms
  • ...and 11 more