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Kendall and Spearman bounds for Chatterjee's rank correlation under positive dependence

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Pith's one-line read For stochastically increasing copulas, Chatterjee's rank correlation satisfies ξ(C) ≤ τ(C), with equality attained by ordinal sums of product copulas.

desk verdict The paper gives sharp, attainable bounds ξ ≤ τ under stochastic increasing and ξ ≤ ρ under joint LTD+RTI, with clean equality cases and directional counterexamples. read the letter →

arxiv 2606.22074 v1 pith:DQAEEC2H submitted 2026-06-20 math.ST stat.TH

Marcus Rockel This is my paper · ORCID
classification math.STstat.TH
keywords copulasChatterjeerankcorrelationKendalltauSpearmanrhostochasticincreasingnesspositivedependencetailmonotonicityorder-violationbounds
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that Chatterjee's rank correlation ξ is bounded above by Kendall's τ whenever the underlying copula is stochastically increasing. This follows from a sharp local inequality that bounds each conditional order-violation probability in τ by the cross-rank variance terms that define ξ. The same approach yields a Spearman bound ξ ≤ ρ under the joint left-tail decreasing and right-tail increasing conditions, with equality only for the independence and comonotonicity copulas. Checkerboard constructions demonstrate that neither condition can be dropped and that the bounds do not reverse direction.

What carries the argument

Sharp order-violation bound for two stochastically ordered distribution functions, which controls conditional order-violation probabilities in τ and ρ by the cross-rank variance terms in ξ.

What would settle it

A single stochastically increasing copula C for which ξ(C) > τ(C), or a copula satisfying both LTD and RTI for which ξ(C) > ρ(C) outside the independence and comonotonicity cases.

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Extended reading notes

Core claim

A sharp order-violation bound for two stochastically ordered distribution functions controls the conditional order-violation probabilities in Kendall's tau by the cross-rank variance functionals of Chatterjee's rank correlation. Consequently, ξ(C) ≤ τ(C) holds for every stochastically increasing copula C, and the bound is attained by ordinal sums of product copulas. Under the weaker joint LTD and RTI conditions, ξ(C) ≤ ρ(C) holds, with equality if and only if C is the independence copula or the comonotonicity copula.

Load-bearing premise

The copula must satisfy stochastic increasingness for the Kendall bound or the joint left-tail decreasing and right-tail increasing properties for the Spearman bound.

Editorial extensions

If this is right

  • The Kendall bound ξ(C) ≤ τ(C) holds for every stochastically increasing copula and is attained by ordinal sums of product copulas.
  • The Spearman bound ξ(C) ≤ ρ(C) holds under joint LTD and RTI, with equality only for the independence and comonotonicity copulas.
  • LTD alone or RTI alone is insufficient to guarantee ξ(C) ≤ ρ(C).
  • Joint LTD and RTI is insufficient to guarantee ξ(C) ≤ τ(C).
  • Both inequalities are one-directional for ξ.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The equality-attaining ordinal sums may serve as test cases when comparing rank correlations numerically.
  • The local order-violation bound could be checked directly on empirical conditional distributions to test applicability of the inequalities to data.
  • The directional nature of the bounds suggests that ξ tends to be smaller than the classical coefficients precisely when positive dependence is present.
  • Checkerboard constructions provide explicit counter-examples that could be used to probe the necessity of the dependence assumptions in other rank-correlation inequalities.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 1 minor

Summary. The paper establishes sharp inequalities relating Chatterjee's rank correlation ξ to Kendall's τ and Spearman's ρ for bivariate copulas under positive dependence. The central technical result is a sharp bound on conditional order-violation probabilities for stochastically ordered distribution functions; this local inequality is integrated to yield ξ(C) ≤ τ(C) for every stochastically increasing copula C, with equality attained by ordinal sums of product copulas. Under the joint LTD and RTI conditions the authors obtain the weaker bound ξ(C) ≤ ρ(C), with equality if and only if C is the independence or comonotonicity copula. Checkerboard constructions demonstrate that the bounds are directional and that the individual tail conditions are insufficient.

Significance. If the derivations hold, the results supply the first sharp comparisons of ξ with the classical rank correlations under explicit positive-dependence hypotheses on copulas. The identification of equality cases and the counter-examples for weaker conditions add precision to the literature on dependence measures. The local order-violation inequality itself appears to be a reusable technical tool.

minor comments (1)
  1. The abstract states that the bound is 'best possible' via ordinal sums of product copulas; a brief explicit construction or reference to the relevant section would help readers locate the equality case immediately.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its technical contributions, and the recommendation to accept. There are no major comments requiring a point-by-point reply.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained from stochastic ordering assumptions

full rationale

The paper establishes a local sharp bound on order-violation probabilities for stochastically ordered distributions, then integrates it under the SI (or joint LTD+RTI) assumptions to obtain the global inequalities ξ ≤ τ and ξ ≤ ρ. The equality-attaining constructions (ordinal sums of product copulas, independence and comonotonicity) are exhibited explicitly and verified directly from the definitions. No parameter fitting, self-referential definitions, load-bearing self-citations, or renaming of known results occurs; the central claims follow from the stated premises and the local inequality without reduction to inputs by construction.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The work rests on standard properties of bivariate copulas and the definitions of the three rank correlations; no free parameters or invented entities are introduced in the abstract.

assumptions (2)
  • domain assumption Stochastically increasing copulas satisfy the positive dependence condition needed to bound order-violation probabilities
    Invoked to obtain ξ(C) ≤ τ(C)
  • domain assumption LTD and RTI conditions on copulas jointly control the relevant tail behaviors for the Spearman comparison
    Invoked to obtain ξ(C) ≤ ρ(C)

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Cite this review

Pith. "Pith review of Kendall and Spearman bounds for Chatterjee's rank correlation under positive dependence." pith.science (2026). https://pith.science/paper/DQAEEC2H

@misc{pith2026260622074,
  author       = {Pith},
  title        = {Pith review of: Kendall and Spearman bounds for Chatterjee's rank correlation under positive dependence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DQAEEC2H}},
  note         = {Machine review of arXiv:2606.22074}
}
abstract

We compare Chatterjee's rank correlation $\xi$ with Kendall's $\tau$ and Spearman's $\rho$ under positive-dependence assumptions on bivariate copulas. Our main technical contribution is a sharp order-violation bound for two stochastically ordered distribution functions. This local inequality controls each conditional order-violation probability appearing in Kendall's tau by the cross-rank variance functionals that determine Chatterjee's rank correlation. As a consequence, we prove the sharp Kendall bound $\xi(C)\leq \tau(C)$ for every stochastically increasing copula $C$. The bound is best possible: ordinal sums of product copulas attain equality. We also prove that the weaker left-tail decreasing (LTD) and right-tail increasing (RTI) conditions jointly imply the Spearman bound $\xi(C)\leq \rho(C)$, with equality if and only if $C$ is either the independence or comonotonicity copula. Finally, checkerboard examples show that LTD or RTI alone does not imply $\xi(C)\leq\rho(C)$, that LTD and RTI together do not imply $\xi(C)\leq\tau(C)$, and that both bounds are directional for $\xi$.

Figures

Figures reproduced from arXiv: 2606.22074 by the authors.

Figure 1
Figure 1. The vertices of the feasible polytope in the proof of Lemma [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Supports of ordinal sums of product copulas for three partitions of [0 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Densities of the four checkerboard copulas used in Example [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

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