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On exact regions between measures of concordance and Chatterjee's rank correlation for lower semilinear copulas

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Pith's one-line read This paper proves exact joint ranges for Kendall's tau, Spearman's rho and footrule, and Chatterjee's xi on lower semilinear copulas, settling a conjecture and showing xi never exceeds the concordance measures.

desk verdict Resolves the tau-rho conjecture with clean arguments, but the claimed "exact" tau-xi region is only a sharp envelope unless a filling argument is supplied. read the letter →

arxiv 2507.23316 v1 pith:2JNSVWI5 submitted 2025-07-31 stat.ME

Sebastian Fuchs , Carsten Limbach , Fabian Schürrer This is my paper · ORCID
classification stat.ME MSC 62H2062H05
keywords concordancemeasuresdirecteddependencefunctionallowersemilinearcopulaMarkovproductKendall'stauSpearman'srhofootrule
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

This paper pins down the exact set of values that four dependence measures can jointly take on the class of lower semilinear copulas, a tractable family built by making a copula linear on each side of the diagonal. It proves the long-conjectured region for Kendall's tau and Spearman's rho, namely $\tau \le \rho \le 1-(1-\tau)^{3/2}$, and derives the analogous exact regions for tau versus Spearman's footrule $\phi$ and for $\phi$ versus rho. It further establishes that Chatterjee's rank correlation $\xi$, a directed measure of how strongly $Y$ depends on $X$, is controlled from above by all three concordance measures and from below by $2\tau^2/(1+\tau)$. A reader should care because these are sharp, complete characterizations of attainable dependence patterns in a widely used copula family, showing precisely how directed dependence is constrained by undirected concordance.

What carries the argument

The load-bearing object is the copula diagonal $\delta(t)=S_\delta(t,t)$ and the chain of shape constraints $\delta(t)\le t\delta'(t)\le 2\delta(t)$ that holds for lower semilinear copulas. Each concordance measure is a simple integral of $\delta$ or $\delta(t)^2/t$, so the region statements reduce to integral inequalities between these functionals; the proofs rewrite differences as integrals of nonnegative terms and apply H\"older (with exponents 3 and 3/2 for tau-rho, 4/3 and 4 for tau-phi) or Jensen to a concave quadratic for the xi lower bound. For xi itself, the key identity is that $\xi(S_\delta)$ equals Spearman's footrule of the Markov product $S_\delta*S_\delta$, whose diagonal is supplied by an explicit formula from the literature, yielding the closed form $\xi(S_\delta)=\tau(S_\delta)-2\int_0^1 (t\delta'(t)-\delta(t))(2\delta(t)-t\delta'(t))/t\,dt$.

What would settle it

Take a single lower semilinear copula whose diagonal is not one of the extremal shapes, compute its Markov product diagonal numerically, and evaluate xi from the definition as a variance ratio; compare with Theorem 3.1 and the claimed region $2\tau^2/(1+\tau)\le\xi\le\tau$. A single mismatch, or one Monte Carlo draw of a random valid diagonal whose (tau, rho) pair falls outside $x\le y\le 1-(1-x)^{3/2}$, would refute the paper's central claim.

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Extended reading notes

Core claim

The paper's central claim is that for every lower semilinear copula $S_\delta$, the quadruple $(\tau, \rho, \phi, \xi)$ must satisfy: $\tau(S_\delta) \le \rho(S_\delta) \le 1-(1-\tau(S_\delta))^{3/2}$; $\tau(S_\delta) \le \phi(S_\delta) \le \tau(S_\delta)^{3/4}$; $\phi(S_\delta)^{4/3} \le \rho(S_\delta) \le 1-(1-\phi(S_\delta))^{3/2}$; and $2\tau(S_\delta)^2/(1+\tau(S_\delta)) \le \xi(S_\delta) \le \tau(S_\delta)$, together with $\xi \le \rho$ and $\xi \le \phi$. In particular, $\xi$ never exceeds any of the three concordance measures. These inequalities are sharp, with equality attained at the copulas with diagonals $u_a$, $l_a$, and the power diagonals $\delta_p(t)=t^p$.

Load-bearing premise

The bounds on Chatterjee's xi are derived from an external formula for the diagonal of the Markov product of a lower semilinear copula; the paper does not re-prove that formula, so if that formula were wrong or inapplicable to some copula in the class, the xi bounds and the tau-xi region would collapse.

Editorial extensions

If this is right

  • The conjecture of Maislinger and Trutschnig is settled: the attainable tau-rho region for lower semilinear copulas is exactly $\{(x,y)\in[0,1]^2: x\le y\le 1-(1-x)^{3/2}\}$, and both bounds are attained.
  • Kendall's tau never exceeds Spearman's footrule, which in turn is at most $\tau^{3/4}$, so the three classical concordance measures stay close on this family.
  • Chatterjee's xi is always smaller than or equal to tau, rho, and phi, with the sharp lower bound $2\tau^2/(1+\tau)\le\xi$; in particular, knowing tau constrains xi to a narrow corridor.
  • The extremal copulas are identified: $S_{u_a}$, $S_{l_a}$, and power diagonals $\delta_p$; the lower tau-xi bound is attained by Marshall-Olkin copulas with equal exponents.
  • The tau-phi and phi-rho regions are convex and compact, and the same holds for tau-xi if continuity of xi in the general case is established.

Reading between the lines

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

  • If these bounds hold, they give a necessary condition for a copula to be lower semilinear: any copula whose dependence values violate one of the inequalities cannot belong to the class, offering a simple diagnostic test for model families.
  • The Markov-product formula for xi could be imported to other copula classes that are closed under the star product, potentially yielding analogous tau-xi regions there; the paper hints at Marshall-Olkin copulas as one such case.
  • Because xi sits strictly below tau, rho, and phi on nearly all of the region, the paper suggests that directed information is more parsimonious than undirected concordance on this family; testing this hypothesis on other families would show whether it is a general principle or a special feature of lower semilinearity.
  • The paper notes that the tau-xi region's convexity is open because continuity of xi is unresolved for general lower semilinear copulas; a constructive proof of continuity via the explicit formula would be a direct next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript studies lower semilinear copulas Sδ and the pairwise relationships among Kendall's τ, Spearman's ρ, Spearman's footrule φ, and Chatterjee's ξ. It proves the τ-ρ conjecture from [18] by establishing τ(Sδ) ≤ ρ(Sδ) ≤ 1 − (1 − τ(Sδ))^{3/2}, determines the τ-φ and φ-ρ regions, derives a closed-form expression for ξ(Sδ) as τ(Sδ) minus a nonnegative integral, proves ξ(Sδ) ≤ τ(Sδ), ρ(Sδ), φ(Sδ), and claims to establish the exact τ-ξ region {(x,y) ∈ [0,1]² : 2x²/(1+x) ≤ y ≤ x}. The main technical arguments are clean Hölder and Jensen inequalities applied to the diagonal function δ and the auxiliary function g.

Significance. If fully justified, the paper gives a parameter-free, complete description of the pairwise regions for four dependence measures on a tractable copula class, resolving an open conjecture and providing novel inequalities for Chatterjee's ξ. The proofs are self-contained and elegant, and the paper explicitly names the external results it relies on. The main caveat is that the exactness (filling) of the τ-ξ region is asserted but not proven; the paper itself flags the missing continuity/convexity ingredients. The inequalities, as opposed to the exact-region equality, are sound and valuable.

major comments (2)
  1. [Section 3.3 / Theorem 3.2] The equality Ω^LSL_{τ,ξ} = {(x,y) ∈ [0,1]² : 2x²/(1+x) ≤ y ≤ x} is not proven. Theorem 3.2 demonstrates the two inequalities and shows that the lower curve is attained by the power diagonals δ_p (Example 2.5) and the upper curve by u_a/l_a (Example 2.1). It does not show that for a fixed τ0 ∈ [0,1] every ξ between 2τ0²/(1+τ0) and τ0 is attained. Section 3.3 explicitly states that continuity of ξ on CLSL is an open problem and that convexity and compactness of the τ-ξ region do not follow directly. Consequently the proof supports a containment result and sharp bounds, not the claimed exact region. This affects the abstract, the caption of Figure 5, and the reported area 3/2 − 2 ln 2. Please add an attainment argument for all intermediate pairs or revise the claims to 'sharp bounds' / 'outer region'.
  2. [Section 2.2, Corollary 2.12] The proof of convexity of Ω^LSL_{τ,φ} is omitted, with the statement that it is analogous to [18, Theorem 5.7]. Since the exactness of the τ-φ region depends on this convexity together with the sharpness of the two boundary curves, the omitted proof is load-bearing. Please include the proof or state precisely which theorem in [18] is being adapted and why it applies to the pair (τ, φ). The same transparency would also help for the convexity facts cited in Section 2.1 for the τ-ρ region.
minor comments (4)
  1. [Theorem 3.1, last display] There is a parenthesis typo in the final integrand: '(2δ(s)) − sδ′(s))' should read '(2δ(s) − sδ′(s))'.
  2. [Proof of Theorem 3.1] The proof invokes '[18, Theorem 4.1]' for the diagonal expression of the Markov product without stating the formula. Since all subsequent ξ results depend on this theorem, please quote the exact statement or the specific formula used, so the applicability to every Sδ ∈ CLSL is transparent.
  3. [Example 2.6] The displayed formulas for τ, ρ, and φ of the Marshall-Olkin copulas appear scrambled by the typesetting (e.g., 'α β α−α β+β 3 α β'). Please check and correct the equations.
  4. [Figure 6] The simulated τ-ρ-φ region in Figure 6 is an illustration, not a theorem; the text should state explicitly that the depicted set is based on simulations and is not part of the proven results.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bounds are proven from the semilinear shape constraints; the only external inputs are independent Markov-product theorems, not fitted or self-referential.

full rationale

The derivation chain is self-contained relative to standard copula facts. The central inequalities (Theorems 2.8, 2.10, 2.13, and 3.2) are obtained directly from the diagonal shape constraints t < t*delta'(t) < 2*delta(t), using integration by parts, Holder's inequality, and Jensen's inequality; no parameter is fitted to the quantities being bounded and no prediction is constructed from an input value. The only external load-bearing ingredients are the representation xi(C) = phi(C^T * C) from [6, 11] and the diagonal formula for Markov products of lower semilinear copulas from [18, Theorem 4.1]. These are independently stated and proved results, not assumptions equivalent to the conclusions drawn; [18] is by different authors, and [6] independently establishes the footrule representation, so the self-citation [11] is not load-bearing. Sharpness is demonstrated by explicit copula diagonals u_a, l_a, and δ_p with independently computed moments. The paper itself flags in Section 3.3 that continuity of xi on all of CLSL remains open and that convexity/compactness of the tau-xi region do not follow directly; this is a technical completeness caveat about filling the region, not circularity. Similarly, the omitted proof of Corollary 2.12 by analogy to [18, Theorem 5.7] is an external-reference gap, not a circular reduction. Consequently, no step reduces by construction to its own inputs or to a fitted constant renamed as a prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the paper introduces no new objects, only new inequalities and regions for the existing family of lower semilinear copulas.

assumptions (4)
  • domain assumption Shape constraints (1) and derivative chain (6) for DLSL diagonals
    Defines the class; used throughout to control g and the sign of the extra term in Theorem 3.1.
  • domain assumption Closed-form tau, rho, phi formulas in (3)
    From [7]; all region proofs start from these integrals.
  • domain assumption xi(C) = phi(C^T * C)
    Representation from [6] and [11], used in (7).
  • domain assumption Diagonal formula for Markov product of LSL copulas from [18, Theorem 4.1]
    Basis for the explicit xi formula in Theorem 3.1.

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Pith. "Pith review of On exact regions between measures of concordance and Chatterjee's rank correlation for lower semilinear copulas." pith.science (2026). https://pith.science/paper/2JNSVWI5

@misc{pith2026250723316,
  author       = {Pith},
  title        = {Pith review of: On exact regions between measures of concordance and Chatterjee's rank correlation for lower semilinear copulas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2JNSVWI5}},
  note         = {Machine review of arXiv:2507.23316}
}
abstract

We explore how the classical concordance measures - Kendall's $\tau$, Spearman's rank correlation $\rho$, and Spearman's footrule $\phi$ - relate to Chatterjee's rank correlation $\xi$ when restricted to lower semilinear copulas. First, we provide a complete characterization of the attainable $\tau$-$\rho$ region for this class, thus resolving the conjecture in [18]. Building on this result, we then derive the exact $\tau$-$\phi$ and $\phi$-$\rho$ regions, obtain a closed-form relationship between $\xi$ and $\tau$, and establish the exact $\tau$-$\xi$ region. In particular, we prove that $\xi$ never exceeds $\tau$, $\rho$, or $\phi$. Our results clarify the relationship between undirected and directed dependence measures and reveal novel insights into the dependence structures that result from lower semilinear copulas.

Figures

Figures reproduced from arXiv: 2507.23316 by the authors.

Figure 1
Figure 1. The copula diagonals δ = ua (left panel) and δ = la (right panel) for a = 1 2 as considered in Example 2.1. We now introduce several additional examples that will be revisited in the subsequent sections. In particular, we construct a diagonal that satisfies, in a piecewise manner, either the lower or the upper bound in (6). Example 2.3. Define the function δ : [0, 1] → [0, 1] by δ(t) :=    8t 2 3 if t ∈… view at source ↗
Figure 2
Figure 2. The copula diagonal considered in Example [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The τ -ρ region ΩLSL τ,ρ According to [18], the τ -ρ region is convex, compact and the lower inequality in Theorem 2.8 is sharp exclusively for copulas of the form Sla with copula diagonal la discussed in Example 2.1. In contrast, as stated in the proof of Theorem 2.8, the upper inequality is sharp for copulas Sua . Remark 2.9. In [10], the authors have established the inequality τ (C) ≤ ρ(C) for copulas C that are … view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The τ -ϕ region ΩLSL τ,ϕ (left panel) and the ϕ-ρ region ΩLSL Φ,ρ (right panel). Remark 2.15. The established τ -ρ, τ -ϕ, and ϕ-ρ regions are valid for lower semilinear copulas Sδ ∈ CLSL. However, for copulas whose copula diagonal δ belongs to the class DLSL but which …
Figure 5
Figure 5. Figure 5: The τ -ξ region ΩLSL τ,ξ Corollary 3.3. Within the class of lower semilinear copula, the identity ξ(Sδ) = τ (Sδ) is attained if and only if, for λ-almost every t ∈ [0, 1], the copula diagonal δ fulfills either t δ′ (t) = δ(t) or t δ′ (t) = 2 δ(t). Corollary 3.3 also ap…
Figure 6
Figure 6. Figure 6: Simulated τ -ρ-ϕ region from two different angles. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the exact region between Chatterjee's rank correlation and Spearman's footrule

    math.ST 2025-09 accept novelty 7.0 of 10

    For any copula, Spearman's footrule cannot exceed the square root of Chatterjee's ξ, and for stochastically increasing copulas the exact attainable region is ξ ≤ ψ ≤ √ξ.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · cited by 1 Pith paper

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