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

Marcus Rockel

REVIEW 3 major objections 4 minor 27 references

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

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The attainable (ξ, ψ) region for stochastically increasing copulas is exactly x ≤ y ≤ √x, with the Fréchet family uniquely tracing the upper curve and ordinal sums of the independence copula tracing the diagonal.

desk verdict A clean, well-executed proof of the upper boundary ψ≤√ξ and the exact (ξ,ψ) region for stochastically increasing copulas; the main caveat is heavy reliance on one external theorem that is not re-derived. read the letter →

arxiv 2509.07232 v1 pith:6S6YPYQP submitted 2025-09-08 math.ST stat.TH

Marcus Rockel This is my paper · ORCID

classification math.STstat.TH MSC 62H20

keywords Chatterjee'srankcorrelationSpearman'sfootrulecopulastochasticallyincreasingattainableregionFréchetMarkovproductordinalsum

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

Chatterjee's rank correlation $\xi$ measures how close $Y$ is to a function of $X$, and Spearman's footrule $\psi$ is a classical rank-association measure; this paper asks which pairs $(\xi,\psi)$ a bivariate copula can produce. It proves that $\psi$ can never exceed $\sqrt{\xi}$, and for any fixed $\xi=x$ the ceiling is reached only by the Fréchet copula with parameter $\alpha=\sqrt{x}$. For the subclass of stochastically increasing copulas—where larger values of one variable make the other stochastically larger—the paper obtains the exact region $x\le y\le\sqrt{x}$: every point is attainable, the diagonal comes from ordinal sums of the independence copula, and the upper curve comes only from the Fréchet family. The full copula region is shown to be convex and closed, with a Jensen-based lower bound and a two-parameter family approaching it, although the exact lower boundary remains open. If correct, these results give a sharp geometric description of how much rank association can accompany a given degree of directed dependence.

What carries the argument

The argument runs through three linked devices. First, the Markov-product identity $\xi(C)=\psi(C^{\top}\ast C)$, where $C^{\top}(u,v)=C(v,u)$ and $(C_1\ast C_2)(u,v)$ is the Markov product; this rewrites Chatterjee's coefficient as Spearman's footrule of a derived copula. Second, Lemma 1.1 characterizes a copula by its partial derivative $h_v(t)=\partial_1 C(t,v)$, subject to $\int_0^1 h_v(t)\,dt=v$ and monotonicity in $v$; in terms of $h$, $\psi$ is a linear functional while $\xi$ is a quadratic functional, turning the region problem into convex optimization over square-integrable functions. Third, for stochastically increasing copulas the diagonal inequality $(C^{\top}\ast C)(v,v)\le C(v,v)$ yields $\xi\le\psi$; the upper boundary then comes from a KKT analysis of the convex program, and the lower boundary of the SI region comes from this diagonal inequality together with continuity of $\xi$ along the convex segment between an ordinal-sum copula and a Fréchet copula.

What would settle it

Take any copula $C$ and compute, on a fine grid of $v\in[0,1]$, the diagonal difference $D(v)=C(v,v)-(C^{\top}\ast C)(v,v)$. If $C$ is stochastically increasing and $D(v)<0$ for even one $v$, then $\xi(C)>\psi(C)$ and the lower boundary of $R_{\mathrm{SI}}$ fails; if for any copula $\psi(C)>\sqrt{\xi(C)}$, the upper-bound theorem fails. The theorems predict no such point exists and that the Fréchet copula is the unique maximizer at each level, so a single numerical counterexample would settle the claim.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 2.4: the attainable $(\xi,\psi)$-region for stochastically increasing copulas is exactly $R_{\mathrm{SI}}=\{(x,y)\in[0,1]^2\mid x\le y\le \sqrt{x}\}$. The lower edge $x=y$ is attained by ordinal sums of the independence copula, and the upper edge $y=\sqrt{x}$ is traced uniquely by the Fréchet copula family $C^{\mathrm{Fr}}_\alpha=(1-\alpha)\Pi+\alpha M$, $\alpha\in[0,1]$. This upper edge is not special to the SI class: Theorem 2.1 states that for every copula $C$, $\psi(C)\le\sqrt{\xi(C)}$, and for fixed $\xi=x$ the only maximizer is the Fréchet copula with $\alpha=\sqrt{x}$. The paper also characterizes equality $\xi=\psi$ inside the SI class by a two-point form of the conditional density, and for the full copula region it proves convexity and closedness, gives a Jensen-derived lower boundary curve, and shows that at $\psi=-1/2$ the smallest possible $\xi$ is $1/2$, attained only by the $2\times2$ checkerboard copula with mass on the off-diagonal squares.

Load-bearing premise

The load-bearing premise is the known diagonal inequality $(C^{\top}\ast C)(v,v)\le C(v,v)$ for stochastically increasing copulas; the full-region lower bound additionally rests on a rearrangement lemma from the author's companion preprint, so either failure would collapse the corresponding boundary.

Editorial extensions

If this is right

Knowing $\xi$ for an SI copula determines $\psi$ up to the band $\xi\le\psi\le\sqrt{\xi}$, and both endpoints of the band are attainable at every level.
For any copula whatsoever, $\psi\le\sqrt{\xi}$ with a unique extremizer at each level: only the Fréchet copula with parameter $\sqrt{x}$ reaches $\psi=\sqrt{x}$ when $\xi=x$.
Equality $\xi=\psi$ is possible at every value from 0 to 1, realized by ordinal sums of the independence copula, so the two measures coincide on a full diagonal of structures.
The corollary $\xi\le\frac{3}{4}\tau+\frac{1}{4}$ for SI copulas connects the square-root band to Kendall's tau.
The full attainable region is convex and closed, so mixtures of copulas with the same $\psi$ fill out intervals of possible $\xi$ values between the two endpoint copulas.

Reading between the lines

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

The paper leaves implicit that the upper ceiling $\psi\le\sqrt{\xi}$ is class-independent: any convex family of copulas containing the Fréchet family will share the same square-root upper boundary, so only the lower boundary distinguishes one subclass from another.
Because the Jensen lower-bound curve is generated by functions that are not true copulas, the exact lower boundary of the full region remains open; a concrete next target is to identify whether the two-parameter family described in Section 3.2, or some refinement of it, actually attains it.
Proposition 2.2 suggests a quantitative stability version: for an SI copula, the gap $\xi-\psi$ should measure the distance from the two-point conditional-density form, which could be turned into a diagnostic for near-functional dependence.
Remark 2.6(c) identifies the attainable region for lower semilinear copulas with the SI region; if the same convexity argument applies to other subclasses that contain the Fréchet family and are closed under mixtures, exact regions for those classes would follow from their equality cases alone.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read Rockel's paper. The stuff worth knowing: Theorem 2.1 gives the upper boundary ψ ≤ √ξ for all copulas, with unique maximizer the Fréchet family. That's a clean result, and the proof is unusually complete for this literature: stationarity, primal/dual feasibility, complementary slackness, positive Hessian, and convexity are all spelled out. I checked the algebra in the KKT system; it works. Theorem 2.4 characterizes the full (ξ,ψ) region over stochastically increasing copulas as x ≤ y ≤ √x, with the diagonal attained by ordinal sums of Π and the upper curve by Fréchet copulas. If true, this is the right extension of the exact-region program from Spearman's ρ to footrule. The paper is honest that the lower boundary of the full copula region is not exact: Jensen gives a bound, a non-copula family C↘_µ is used to evaluate that bound, and a two-parameter copula family gets close numerically. That is a real limitation but stated openly, not a hidden flaw.

The main soft spot is exactly the one flagged in the stress test: the SI region hinges on [24, Thm. 4.2], the pointwise Markov-product inequality (C⊤∗C)(v,v) ≤ C(v,v). The paper invokes it twice without proof. I haven't gone through [24] myself, so I can't certify it from the paper alone. But it is a published theorem in J. Math. Anal. Appl. from 2021, not a private communication, and the usage looks faithful. If I were refereeing, I'd ask the author to state the result explicitly in a self-contained way or at least quote it in full. This is a prop, not a reason to reject. The paper also leans on the author's own preprint [4] for the copula-derivative characterization; that lemma is plausible and was presumably already vetted for the ρ-paper, but a referee may want it stated rather than cited.

Theorem 3.4 (minimal ξ = 1/2 when ψ = −1/2, uniquely via checkerboard) is a nice bonus. It uses external results from [13] and [23], but those are also legitimate published checks.

Bottom line: solid mathematical paper, well executed, clear scope. The exact SI region is the kind of result people will cite. It deserves a serious referee. My only real request is transparency about the cited external inequalities.

Referee Report

3 major / 4 minor

Summary. The paper studies the attainable joint range of Chatterjee's rank correlation ξ and Spearman's footrule ψ over bivariate copulas. The main results are: (i) Theorem 2.1 proves that for each x∈[0,1] the maximum possible ψ given ξ=x is √x, uniquely attained by the Fréchet copula C^Fr_α with α=√x; the proof uses a convex optimization reformulation over partial derivatives and verifies the KKT conditions as well as a quadratic-growth second-order condition. (ii) Theorem 2.4 characterizes the exact (ξ,ψ)-region for stochastically increasing copulas as R_SI = {(x,y): x≤y≤√x}, with the lower boundary attained by ordinal sums of the independence copula and the upper boundary uniquely by the Fréchet family. (iii) Section 3 provides a lower bound for the minimal ψ given ξ, based on Jensen's inequality, proves that at ψ=-1/2 the minimal ξ is 1/2 (uniquely attained by a checkerboard copula), and constructs a two-parameter copula family that approximately realizes the lower boundary. The paper is clearly written and explicitly acknowledges that the lower boundary for all copulas is not exactly characterized.

Significance. If correct, the upper-bound theorem and the exact SI region are substantive, elegant results: they give a sharp, simple geometric description of the joint range of two popular dependence measures, and the uniqueness of the Fréchet maximizer is non-obvious. The KKT proof of Theorem 2.1 is detailed and rigorous, including stationarity, primal/dual feasibility, complementarity, positive-definite Hessian, and convexity-based global minimality. The paper also gives a clean construction (ordinal sums of Π) attaining the full diagonal in the SI region, and it is honest about the fact that the global lower boundary is only bounded, not exactly reached. A notable caveat is that the SI-region result rests on the external pointwise inequality (C^T*C)(v,v) ≤ C(v,v) for stochastically increasing copulas, cited from [24, Thm. 4.2]; the manuscript does not re-derive this inequality, so the exactness of R_SI is contingent on that published theorem. Additionally, several supporting results in Section 3 are quoted from the author's own preprints [4] and [23], which are not yet peer-reviewed.

major comments (3)

[Section 3.3, proof of Theorem 3.3 (convexity of R)] The proof that the full attainable region R is convex is incomplete. The argument mixes two lower-boundary copulas C0 and C1 and observes that x_λ := ξ(C_λ) ≤ (1−λ)x0+λx1 and y_λ := ψ(C_λ) = (1−λ)φ(x0)+λφ(x1). However, membership of (x_λ,y_λ) in R gives only y_λ ≥ φ(x_λ), and since x_λ may be strictly smaller than the weighted average, this does not imply the convexity inequality φ((1−λ)x0+λx1) ≤ (1−λ)φ(x0)+λφ(x1). The subsequent sentence that the point 'lies weakly below the straight line' does not establish that it lies above the lower boundary. Moreover, the assertion that 'R is only limited by the upper and lower boundaries' requires a proof that vertical slices are intervals; the paper only establishes that horizontal slices R_y are intervals. The closure proof depends on the same convexity claim. Please provide a complete proof or weaken the statement of Theorem 3.3 by omitting the convexity/closedness assertion, which is not needed for the inclusion (25).
[Theorem 3.3, statement (25)] The displayed inclusion in Theorem 3.3 is not well-defined for y>0. The parameter µ(y) is introduced as the unique solution in [0,2] to the cubic equation, but as the proof itself notes, the equation has such a solution only for y∈[−1/2,0], because ψ(C↘_µ) ranges from 0 to −1/2 on that interval. For y>0, the expression ξ(C↘_µ(y)) is therefore undefined, yet the set in (25) ranges over y∈[−1/2,1]. Please restrict the lower-bound inequality to y≤0 and state separately that for y>0 the only bound used is y≤√x (equivalently x≥y^2).
[Section 3.3, reduction to SD copulas and external dependencies] The proof of Theorem 3.3 relies on [4, Prop. 2.4 & Cor. 2.5] (the existence of an SD rearrangement preserving ξ and not increasing ψ) and on [2, Cor. 3.6] (continuity of ξ on C_SD). Both are results from unpublished preprints by the author and coauthors. Since this reduction is load-bearing for the lower-bound inclusion (25), the manuscript should either state these results explicitly with proofs or cite published versions; otherwise a reader cannot verify the key step that inf{ψ(C): ξ(C)=x} equals the infimum restricted to SD copulas. If the results are correct, this is a strength, but the current manuscript makes the verification difficult.

minor comments (4)

[Section 3.2, Eq. (33)] The formula α(µ) = 1/2 − 2/5 µ for µ>2 yields negative values (e.g., µ=2.5 gives α=−0.5), contradicting the standing assumption α∈[0,0.5). This makes the path C_µ undefined for µ>2. If the intention was to continue from the point (0.3,0.5) toward (0.5,0.5) along β=0.5, a different expression (possibly 1/2 − 2/(5µ)) is needed.
[Proof of Theorem 3.2, pointwise problem (22)] The displayed objective 'minimize µvh 1 + vh 2 1 + (1−v)h 2 2' is typographically confusing; it should read µ v h_1 + v h_1^2 + (1−v) h_2^2. Please clarify the notation.
[Introduction and references] Several load-bearing results are cited from the author's preprints [4] and [23], as well as from other very recent preprints ([1], [2], [14]). It would help the reader if the introduction or a remark explicitly stated which external results are required for each main theorem and which of these are not yet published.
[Figure 1 caption] The caption refers to a 'solid line from Π to C#' and a 'dotted line from Π to C#' as lower-bound curves. In black-and-white print these two curves may be difficult to distinguish; please use more distinct line styles or colors and add explicit labels in the figure itself.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the central upper-bound and SI-region results come from explicit optimization and external theorems; only minor self-citations appear in auxiliary lemmas.

full rationale

The derivation chain is not circular. The headline upper bound (Theorem 2.1) is obtained by solving a convex optimization problem in the partial derivative h, with objective determined by ψ and constraint ξ≤x; the candidate h* is explicitly checked against the KKT conditions and is exactly the derivative of the Fréchet copula with α=√x. No parameter is fitted to the target and then renamed as a prediction; the equality ψ=√ξ follows from the explicit optimizer. The SI region (Theorem 2.4) uses the external Markov-product inequality (C^T*C)(v,v)≤C(v,v) from [24, Thm. 4.2] and [24, Thm. 5.1] for idempotence of ordinal sums of Π; these are genuine external supports, not self-citations. The diagonal x=y is attained by ordinal sums of Π and the interior is filled by convexity plus continuity of ξ along the explicit quadratic path (15), so the region is not assumed into existence. The lower-boundary part is also handled honestly: C↘_μ is explicitly said not to be a true copula, and the α=3/5β choice in Section 3.2 is explicitly described as numerically motivated and not claimed as a derived law, so it does not disguise a fit as a prediction. The only mild concern is that the optimization framework, Lemma 1.1, and a few auxiliary facts are cited from the author’s own preprint [4], and Theorem 3.4’s key inequality at ψ=−1/2 is imported from the same author’s preprint [23] together with external [13, Thm. 3.2] and [1, Thm. 2.2 & Cor. 2.3]. These are technical lemmas rather than the target result, and the central ξ≤ψ≤√ξ characterization does not reduce to them by construction. Accordingly, the paper has no significant circularity; the score of 2 reflects minor self-citations that are not load-bearing for the main claim.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

All central theorems are derived in the paper from standard copula theory and convex optimization. The paper depends on cited external theorems (Siburg-Strothmann, Fuchs-Mccord, Rockel, Bonnans-Shapiro) and on the author's own prior preprint [4] for a structural lemma and an SD-rearrangement step. No new physical entities are invoked. The one hand-chosen numerical constant α=3/5β affects only the heuristic lower-bound path in Section 3.2.

free parameters (1)

Heuristic α/β ratio in C_µ path = α(µ)=3/20 µ for µ≤2, β(µ)=µ/4
Chosen by hand from the shapes of discrete numerical minimizers (Figure 2) to define the heuristic one-parameter path C_µ in Section 3.2. It is not used in the main theorems and is explicitly acknowledged as not proven optimal.

assumptions (6)

domain assumption Lemma 1.1 characterization of copulas via partial derivatives (from [4])
Used to reformulate optimization over h in Theorems 2.1 and 3.2.
domain assumption [24, Thm. 4.2]: for SI copulas, (C⊤∗C)(v,v) ≤ C(v,v) for all v
Basis for the lower bound ξ≤ψ in Prop. 2.2 and Thm. 2.4.
domain assumption [24, Thm. 5.1]: ordinal sums of Π are SI, symmetric, and idempotent
Used to attain the diagonal ξ=ψ in Thm. 2.4.
ad hoc to paper [4, Prop. 2.4 & Cor. 2.5]: every copula admits an SD rearrangement with ψ(C′)≤ψ(C) and ξ(C′)=ξ(C)
Reduces lower-bound minimization to SD copulas in Thm. 3.3; relies on a preprint by the same author.
standard math [5, Thm. 3.63]: quadratic growth condition for KKT sufficiency in Banach spaces
Used in Lemma A.2 for Theorems 2.1 and 3.2.
domain assumption [13, Thm. 3.2] and [23, Thm. 4.1]: characterize ψ=-0.5 copulas as 2×2 off-diagonal checkerboards and give lower bound on ξ for checkerboards
Used in Theorem 3.4 to identify the unique minimizer C#.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the exact region between Chatterjee's rank correlation and Spearman's footrule." pith.science (2026). https://pith.science/paper/6S6YPYQP

@misc{pith2026250907232,
  author       = {Pith},
  title        = {Pith review of: On the exact region between Chatterjee's rank correlation and Spearman's footrule},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6S6YPYQP}},
  note         = {Machine review of arXiv:2509.07232}
}

abstract

Chatterjee's rank correlation $\xi$ has emerged as a popular measure quantifying the strength of directed functional dependence between random variables $X$ and $Y$. If $X$ and $Y$ are continuous, $\xi$ equals Spearman's footrule~$\psi$ for the Markov product of the copula induced by $(X,Y)$ and its transpose. We analyze the relationship between these two measures more in depth by studying the attainable region of possible pairs $(\xi, \psi)$ over all bivariate copulas. In particular, we show that for given $\xi$, the maximal possible value of $\psi$ is uniquely attained by a Fr\'echet copula. As a by-product of this and a known result for Markov products of copulas, we obtain that $\xi\le\psi\le \sqrt{\xi}$ characterizes the exact region of stochastically increasing copulas. Regarding the minimal possible value of $\psi$ for given $\xi$, we give a lower bound based on Jensen's inequality and construct a two-parameter copula family that comes comparably close.

Figures

Figures reproduced from arXiv: 2509.07232 by the authors.

**Figure 1.** The attainable (ξ, ψ)-region for all copulas on the left and the (ξ, ψ)-region for stochastically increasing copulas on the right. Π(u, v) := uv, M(u, v) := min{u, v} and W(u, v) := max{u + v − 1, 0}, u, v ∈ [0, 1], denote the independence, upper and lower Fr´echet copulas, respectively, C# the 2 × 2-checkerboard copula with zero mass on the main diagonal, which minimizes ξ over all copulas with ψ = −0.5, and C ↘ … view at source ↗

**Figure 2.** Density plots for discrete minimizers of the convex optimization problem (29) for different [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** A visualization of Cα,β for (α, β) = (0.2, 0.3) (left) and (α, β) = (0.3, 0.5) (right). The parameters describe the shapes of rectangles in the lower-left and upper-right corners, which together with a connecting band form the shape Hα,β where the copula should have a density of zero. The parameter transform bends this zero-density area to ensure the uniformity of the marginals, hence Hα,β does not exactly match the… view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 21 canonical work pages

[24]

Stochastic monotonicity and the Markov product for copulas.J

Karl Friedrich Siburg and Christopher Strothmann. Stochastic monotonicity and the Markov product for copulas.J. Math. Anal. Appl., 503(2):14, 2021

work page 2021
[4]

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

Jonathan Ansari and Marcus Rockel. The exact region and an inequality between chatterjee’s and spearman’s rank correlations.arXiv preprint arXiv:2506.15897, 2025

work page Pith review arXiv 2025
[13]

On the lower bound of Spearman’s footrule.Depend

Sebastian Fuchs and Yann Mccord. On the lower bound of Spearman’s footrule.Depend. Model., 7:126–132, 2019

work page 2019
[23]

Measures of association for approximating copulas.arXiv preprint arXiv:2505.08045, 2025

Marcus Rockel. Measures of association for approximating copulas.arXiv preprint arXiv:2505.08045, 2025

arXiv 2025
[1]

A simple extension of azadkia and chatterjee’s rank correlation to a vector of endogenous variables.arXiv preprint arXiv:2212.01621, 2022

Jonathan Ansari and Sebastian Fuchs. A simple extension of azadkia and chatterjee’s rank correlation to a vector of endogenous variables.arXiv preprint arXiv:2212.01621, 2022

arXiv 2022
[2]

On continuity of chatterjee’s rank correlation and related dependence measures.arXiv preprint arXiv:2503.11390, 2025

Jonathan Ansari and Sebastian Fuchs. On continuity of chatterjee’s rank correlation and related dependence measures.arXiv preprint arXiv:2503.11390, 2025

arXiv 2025
[3]

Dependence properties of bivariate copula families

Jonathan Ansari and Marcus Rockel. Dependence properties of bivariate copula families. Depend. Model., 12:36, 2024

2024
[5]

Fr´ ed´ eric Bonnans and Alexander Shapiro.Perturbation Analysis of Optimization Problems

J. Fr´ ed´ eric Bonnans and Alexander Shapiro.Perturbation Analysis of Optimization Problems. Springer Ser. Oper. Res. New York, NY: Springer, 2000. 16

work page 2000

Show all 27 references

[6]

On the exact region determined by Spearman’s footrule and Gini’s gamma.J

Damjana Kokol Bukovˇ sek and Blaˇ z Mojˇ skerc. On the exact region determined by Spearman’s footrule and Gini’s gamma.J. Comput. Appl. Math., 410:13, 2022. Id/No 114212

2022
[7]

A new coefficient of correlation.J

Sourav Chatterjee. A new coefficient of correlation.J. Amer. Statist. Assoc., 116(536):2009– 2022, 2021

2009
[8]

Darsow, Bao Nguyen, and Elwood T

William F. Darsow, Bao Nguyen, and Elwood T. Olsen. Copulas and Markov processes.Ill. J. Math., 36(4):600–642
[9]

Siburg, and Pavel A

Holger Dette, Karl F. Siburg, and Pavel A. Stoimenov. A copula-based non-parametric mea- sure of regression dependence.Scand. J. Stat., 40(1):21–41, 2013

2013
[10]

On the approximation of copulas via shuffles of Min.Statist

Fabrizio Durante and Juan Fern´ andez S´ anchez. On the approximation of copulas via shuffles of Min.Statist. Probab. Lett., 82(10):1761–1767, 2012

2012
[11]

Boca Raton, FL: CRC Press, 2016

Fabrizio Durante and Carlo Sempi.Principles of Copula Theory. Boca Raton, FL: CRC Press, 2016

2016
[12]

Quantifying directed dependence via dimension reduction.J

Sebastian Fuchs. Quantifying directed dependence via dimension reduction.J. Multivariate Anal., 201:21, 2024

2024
[14]

On exact regions between measures of concordance and chatterjee’s rank correlation for lower semilinear copulas.arXiv preprint arXiv:2507.23316, 2025

Sebastian Fuchs, Carsten Limbach, and Fabian Sch¨ urrer. On exact regions between measures of concordance and chatterjee’s rank correlation for lower semilinear copulas.arXiv preprint arXiv:2507.23316, 2025

2025 arXiv
[15]

Spearman’s footrule and Gini’s gamma: a review with complements.J

Christian Genest, Johanna Neˇ slehov´ a, and Noomen Ben Ghorbal. Spearman’s footrule and Gini’s gamma: a review with complements.J. Nonparametr. Stat., 22(8):937–954, 2010

2010
[16]

A modern treatment of the theory of functions of a real variable

Edwin Hewitt and Karl Stromberg.Real and abstract analysis. A modern treatment of the theory of functions of a real variable. 3rd printing, volume 25 ofGrad. Texts Math.Springer, Cham, 1975

1975
[17]

On the exact regions determined by Kendall’s tau and other concordance measures.Mediterr

Damjana Kokol Bukovˇ sek and Nik Stopar. On the exact regions determined by Kendall’s tau and other concordance measures.Mediterr. J. Math., 20(3):16, 2023

2023
[18]

On the exact region determined by Spearman’s rho and Spearman’s footrule.J

Damjana Kokol Bukovˇ sek and Nik Stopar. On the exact region determined by Spearman’s rho and Spearman’s footrule.J. Comput. Appl. Math., 437:23, 2024

2024
[19]

Spearman’s footrule and Gini’s gamma: local bounds for bivariate copulas and the exact region with respect to Blomqvist’s beta.J

Damjana Kokol Bukovˇ sek, Tomaˇ z Koˇ sir, Blaˇ z Mojˇ skerc, and Matjaˇ z Omladiˇ c. Spearman’s footrule and Gini’s gamma: local bounds for bivariate copulas and the exact region with respect to Blomqvist’s beta.J. Comput. Appl. Math., 390:24, 2021. Id/No 113385

2021
[20]

Linear algebra, Galois theory, representation theory, group extensions and Schur multiplier

Ramji Lal.Algebra 2. Linear algebra, Galois theory, representation theory, group extensions and Schur multiplier. Infosys Sci. Found. Ser. Singapore: Springer, 2017

2017
[21]

Nelsen.An Introduction to Copulas

Roger B. Nelsen.An Introduction to Copulas. 2nd ed.New York, NY: Springer, 2006

2006
[22]

Peˇ cari´ c, Frank Proschan, and Y

Josip E. Peˇ cari´ c, Frank Proschan, and Y. L. Tong.Convex functions, partial orderings, and statistical applications, volume 187 ofMath. Sci. Eng.Boston, MA etc.: Academic Press, 1992

1992
[25]

M. Sklar. Fonctions de r´ epartition ` andimensions et leur marges.Publ. Inst. Stat. Univ. Paris, 8:229–231, 1960

1960
[26]

On a strong metric on the space of copulas and its induced dependence measure.J

Wolfgang Trutschnig. On a strong metric on the space of copulas and its induced dependence measure.J. Math. Anal. Appl., 384(2):690–705, 2011

2011
[27]

Revisiting the region deter- mined by spearman’sρand spearman’s footruleϕ.Journal of Computational and Applied Mathematics, 457:116259, 2025

Marco Tschimpke, Manuela Schreyer, and Wolfgang Trutschnig. Revisiting the region deter- mined by spearman’sρand spearman’s footruleϕ.Journal of Computational and Applied Mathematics, 457:116259, 2025. 17

2025

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [24]

Stochastic monotonicity and the Markov product for copulas.J

Karl Friedrich Siburg and Christopher Strothmann. Stochastic monotonicity and the Markov product for copulas.J. Math. Anal. Appl., 503(2):14, 2021

work page 2021

[2] [4]

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

Jonathan Ansari and Marcus Rockel. The exact region and an inequality between chatterjee’s and spearman’s rank correlations.arXiv preprint arXiv:2506.15897, 2025

work page Pith review arXiv 2025

[3] [13]

On the lower bound of Spearman’s footrule.Depend

Sebastian Fuchs and Yann Mccord. On the lower bound of Spearman’s footrule.Depend. Model., 7:126–132, 2019

work page 2019

[4] [23]

Measures of association for approximating copulas.arXiv preprint arXiv:2505.08045, 2025

Marcus Rockel. Measures of association for approximating copulas.arXiv preprint arXiv:2505.08045, 2025

arXiv 2025

[5] [1]

A simple extension of azadkia and chatterjee’s rank correlation to a vector of endogenous variables.arXiv preprint arXiv:2212.01621, 2022

Jonathan Ansari and Sebastian Fuchs. A simple extension of azadkia and chatterjee’s rank correlation to a vector of endogenous variables.arXiv preprint arXiv:2212.01621, 2022

arXiv 2022

[6] [2]

On continuity of chatterjee’s rank correlation and related dependence measures.arXiv preprint arXiv:2503.11390, 2025

Jonathan Ansari and Sebastian Fuchs. On continuity of chatterjee’s rank correlation and related dependence measures.arXiv preprint arXiv:2503.11390, 2025

arXiv 2025

[7] [3]

Dependence properties of bivariate copula families

Jonathan Ansari and Marcus Rockel. Dependence properties of bivariate copula families. Depend. Model., 12:36, 2024

2024

[8] [5]

Fr´ ed´ eric Bonnans and Alexander Shapiro.Perturbation Analysis of Optimization Problems

J. Fr´ ed´ eric Bonnans and Alexander Shapiro.Perturbation Analysis of Optimization Problems. Springer Ser. Oper. Res. New York, NY: Springer, 2000. 16

work page 2000