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arxiv: 2506.15897 · v3 · pith:37VE5GXZnew · submitted 2025-06-18 · 🧮 math.ST · math.PR· stat.TH

The exact region and an inequality between Chatterjee's and Spearman's rank correlations

Jonathan Ansari , Marcus Rockel This is my paper

Pith reviewed 2026-05-21 23:37 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.TH
keywords Chatterjee rank correlationSpearman rhocopulasdependence measuresattainable regionrank correlationsstochastic monotonicity
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The pith

The attainable pairs of Chatterjee's ξ and Spearman's ρ form a convex set whose boundary is traced by a new family of asymmetric copulas with diagonal band structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper maps the exact set of possible simultaneous values for Chatterjee's rank correlation ξ and Spearman's ρ. A reader would care because ξ quantifies functional dependence while ρ captures concordance, and the two measures routinely disagree in strength on the same data. The work proves the joint region is convex, characterizes its boundary through a novel family of absolutely continuous asymmetric copulas, shows ξ ≤ |ρ| whenever the dependence is stochastically monotone, and pins the largest possible gap ρ − ξ at exactly 0.4, which suggests that √ξ provides a closer match to the scale of ρ.

Core claim

The set of attainable pairs (ξ(X,Y), ρ(X,Y)) is a convex subset of [0,1] × [-1,1]. Its boundary is characterized by a novel family of absolutely continuous, asymmetric copulas having a diagonal band structure. Moreover, ξ(X,Y) ≤ |ρ(X,Y)| holds whenever Y is stochastically increasing or decreasing in X, and the maximal value of ρ(X,Y) − ξ(X,Y) equals 0.4. These facts are established by formulating the problem as a convex optimization task subject to equality and inequality constraints together with ordering properties of the two correlations.

What carries the argument

The ξ-ρ-region, the closed convex set of all attainable pairs (ξ(X,Y), ρ(X,Y)), whose boundary is realized by the new family of absolutely continuous asymmetric copulas with diagonal band structure.

If this is right

  • The attainable set of pairs (ξ, ρ) is convex.
  • The boundary of the set is attained by absolutely continuous asymmetric copulas with diagonal band structure.
  • ξ(X,Y) ≤ |ρ(X,Y)| whenever Y is stochastically monotone in X.
  • The supremum of ρ(X,Y) − ξ(X,Y) equals 0.4.
  • For positive dependence, √ξ aligns more closely with the numerical scale of ρ than ξ itself does.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The boundary copulas supply explicit extremal examples that can be used to benchmark numerical estimators of both correlations.
  • Similar convex-optimization characterizations could be attempted for other pairs of rank correlations such as Kendall's tau paired with ξ.
  • Empirical checks of whether observed (ξ̂, ρ̂) pairs fall inside the region offer a simple consistency diagnostic for real data.

Load-bearing premise

The convex optimization problem under the imposed equality and inequality constraints correctly locates the global boundary of the attainable set for every possible joint distribution.

What would settle it

A pair of random variables X and Y for which ρ(X,Y) − ξ(X,Y) exceeds 0.4, or for which the pair (ξ, ρ) lies strictly outside the claimed convex set, would contradict the stated maximal difference and boundary characterization.

Figures

Figures reproduced from arXiv: 2506.15897 by Jonathan Ansari, Marcus Rockel.

Figure 1
Figure 1. Figure 1: The attainable region R of the tuple (ξ(C), ρ(C)) for bivariate copulas C ∈ C ; see The￾orem 1.1. Stochastically increasing (decreasing) copulas are located in the upper (lower) scattered area; see Theorem 1.3. The upper and lower boundary of the region is described by the copula family (Cb)b∈R\{0} with limiting cases Π for b → 0 , M for b → ∞ , and W for b → −∞ ; see Theorem 3.2 and Remark 3.3(b). In cont… view at source ↗
Figure 2
Figure 2. Figure 2: The class of bivariate copulas with level sets of Spearman’s [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Furthermore, for b < 0, define Cb(u, v) := v − C−b(1 − u, v), (u, v) ∈ [0, 1]2 ., Due to the following theorem, which is the basis for Theorem 1.1, the functions Cb , b ∈ R\{0} , are copulas that solve the optimization problem (13). More precisely, for each c ∈ (0, 1), we determine the copula parameter b and show that Cb is the unique solution to (13). The lengthy proof is based on solving the convex Optim… view at source ↗
Figure 3
Figure 3. Figure 3: The density (left) and derivative (right) of the copula [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

The rank correlation \xi(X,Y), recently established by Sourav Chatterjee and already popular in the statistics literature, takes values in [0,1], where 0 characterizes independence of X and Y, and 1 characterizes perfect dependence of Y on X. Unlike concordance measures such as Spearman's \rho, which capture the degree of positive or negative dependence, \xi quantifies the strength of functional dependence. In this paper, we study the attainable set of pairs (\xi(X,Y),\rho(X,Y)). The resulting {\xi}-\r{ho}-region is a convex set whose boundary is characterized by a novel family of absolutely continuous, asymmetric copulas having a diagonal band structure. Moreover, we prove that \xi(X,Y)\leq|\rho}(X,Y)| whenever Y is stochastically increasing or decreasing in X, and we identify the maximal difference \rho(X,Y)-\xi(X,Y) as exactly 0.4. Our proofs rely on a convex optimization problem under various equality and inequality constraints, as well as on ordering properties for \xi and \rho. Our results contribute to a better understanding of Chatterjee's rank correlation, which typically yields substantially smaller values than Spearman's \rho when quantifying positive dependencies. In particular, when interpreting the values of Chatterjee's rank correlation on the scale of \rho, the quantity \sqrt{\xi} appears to be more appropriate.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript determines the exact attainable region for pairs (ξ(X,Y), ρ(X,Y)), where ξ denotes Chatterjee's rank correlation and ρ denotes Spearman's rank correlation. It claims that this region is a convex set whose boundary is attained by a novel family of absolutely continuous asymmetric copulas with diagonal band structure. The paper also proves that ξ(X,Y) ≤ |ρ(X,Y)| whenever Y is stochastically increasing or decreasing in X, and identifies the maximal difference ρ(X,Y) − ξ(X,Y) as exactly 0.4. All results are obtained by reducing the problem to a convex optimization program subject to equality and inequality constraints derived from the definitions of ξ, ρ, and copula ordering properties.

Significance. If the central claims hold, the work supplies a precise geometric description of the relationship between Chatterjee's functional-dependence measure and the classical concordance measure, together with an explicit extremal copula family and the sharp constant 0.4. These findings are useful for calibrating the scale of ξ against ρ and for understanding why ξ typically returns smaller values than ρ under positive dependence. The explicit construction of the boundary copulas and the reduction to a convex program constitute a technically substantive contribution, provided the optimization is shown to be globally exhaustive.

major comments (1)
  1. [§4] §4, optimization problem (4.1)–(4.5): the claim that the computed convex set is exactly the attainable (ξ,ρ)-region rests on the assertion that the chosen parameterization (asymmetric diagonal-band copulas) together with the listed equality/inequality constraints exhausts all possible joint distributions. No explicit verification is given that every candidate extremal copula outside this family yields a point strictly inside the computed boundary; if such a copula exists, both the region description and the numerical value 0.4 would be incorrect.
minor comments (2)
  1. [§3] The notation for the diagonal-band copula family is introduced without a compact symbolic definition; a single displayed equation collecting the density or distribution function would improve readability.
  2. [Figure 2] Figure 2 (boundary plot) lacks axis labels indicating the precise numerical range of the maximal gap; adding the value 0.4 as a marked point would clarify the geometric claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the single major comment below, providing a clarification of our approach while agreeing that an explicit statement on exhaustiveness will strengthen the presentation.

read point-by-point responses

  1. Referee: [§4] §4, optimization problem (4.1)–(4.5): the claim that the computed convex set is exactly the attainable (ξ,ρ)-region rests on the assertion that the chosen parameterization (asymmetric diagonal-band copulas) together with the listed equality/inequality constraints exhausts all possible joint distributions. No explicit verification is given that every candidate extremal copula outside this family yields a point strictly inside the computed boundary; if such a copula exists, both the region description and the numerical value 0.4 would be incorrect.

    Authors: The equality and inequality constraints in (4.1)–(4.5) are necessary conditions satisfied by every copula, because they are obtained directly from the integral definitions of ξ and ρ together with the monotonicity and ordering properties of copulas. The asymmetric diagonal-band family is not claimed to be the only possible family; it is used because it is sufficiently flexible to attain every point on the boundary of the feasible set defined by those constraints. Consequently, no copula (inside or outside the family) can produce a pair (ξ, ρ) lying outside the computed convex set, as that would violate a necessary constraint. Interior points are attainable by convex combinations of the boundary copulas or by standard families such as the independence and Fréchet–Hoeffding bounds. We will revise the manuscript to add an explicit paragraph after the statement of the optimization problem that spells out this necessity argument and confirms that the boundary is sharp because it is attained by the proposed family. revision: yes

Circularity Check

0 steps flagged

No significant circularity; boundary from external convex optimization and ordering properties

full rationale

The derivation relies on formulating and solving a convex optimization problem subject to equality and inequality constraints on copulas, combined with standard ordering properties of ξ and ρ. The attainable region and the value 0.4 are outputs of this optimization rather than inputs redefined as results. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided abstract or description. The approach is self-contained against external benchmarks (convex programming and rank-correlation inequalities), consistent with a low circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, preventing a complete audit. The work relies on standard properties of copulas and rank correlations plus a convex optimization formulation; no free parameters or invented physical entities are evident from the abstract.

axioms (1)
  • standard math Standard properties of copulas and rank correlations hold, including the representation of dependence via copulas and the definitions of ξ and ρ.
    The paper invokes these background facts to set up the optimization problem and ordering arguments.

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Reference graph

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