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arxiv: 2605.13522 · v3 · pith:SZ7TP2DInew · submitted 2026-05-13 · 🧮 math.ST · stat.TH

Dependence functions based on Chatterjee's rank correlation

Carsten Limbach This is my paper

Pith reviewed 2026-05-20 21:17 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords Chatterjee's rank correlationdependence functionsMarkov productstochastic dependencefunctional dependencerank correlationdirected dependence
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The pith

Chatterjee's ξ-coefficient extends to dependence functions via the Markov product that also quantify concentration near the diagonal.

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

This paper aims to reinterpret Chatterjee's ξ-coefficient for functional dependence between Y and predictors X by examining the Markov product of Y with a conditionally independent copy Y' given X. The authors introduce two dependence functions, φ and κ, built on this product. These functions assess not only how well Y can be expressed as a function of X but also how closely the paired values cluster near the diagonal in their joint distribution. A sympathetic reader would value this because it replaces a single scalar with objects that expose more structure in directed dependence relations.

Core claim

By analyzing the Markov product (Y, Y') where Y' is a copy of Y that is conditionally independent of Y given X, the paper defines dependence functions φ_{(Y,X)} and κ_{(Y,X)} that extend the original ξ-coefficient. These functions deliver a geometric interpretation of the Markov product and measure the strength of concentration near the diagonal in addition to the degree of functional dependence.

What carries the argument

The Markov product (Y, Y'), the pair formed by the response and its copy that is conditionally independent given the predictors, which underpins the geometric and distributional reinterpretation.

If this is right

  • The dependence functions quantify both functional dependence of Y on X and the degree of concentration of the Markov product near the diagonal.
  • The construction supplies a geometric interpretation of the Markov product for studying directed stochastic dependence.
  • Analysts obtain a richer object than the scalar ξ-coefficient that reveals additional aspects of the relationship structure.

Where Pith is reading between the lines

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

  • These functions might serve as the basis for new visualization methods that plot dependence strength across different regions of the predictor space.
  • They could be compared against other dependence measures to identify cases where concentration near the diagonal captures features missed by scalar coefficients.
  • The approach might support the development of tests that detect specific forms of dependence by examining deviations in the concentration component.

Load-bearing premise

The Markov product of Y with its conditionally independent copy given X supplies a consistent geometric and distributional reinterpretation that extends the original coefficient without inconsistencies.

What would settle it

A direct computation of the proposed functions on a simple case such as perfect functional dependence where Y equals a function of X, checking whether they reach their maximum value of one while showing full concentration on the diagonal.

Figures

Figures reproduced from arXiv: 2605.13522 by Carsten Limbach.

Figure 1
Figure 1. Figure 1: From left to right: the random vector (Y, X) associated with a Gaussian copula (2.8) with parameter 0.65, a Marshall–Olkin copula (2.9) with parameters (1, 1/3), and a jump copula with parameter 1 (see Example 2.10), based on a sample size of n = 2500. We have the following characterizations of independence and perfect dependence: ϕ(Y,X)(t) = 2t − t 2 for all t ∈ [0, 1] if and only if X ⊥ Y, whereas ϕ(Y,X)… view at source ↗
Figure 2
Figure 2. Figure 2: The dependence functions ϕ(Y,X) and κ for a random vector (Y, X) associated with a Gaussian copula (2.8) with parameter ρ = 0.65 (red), a Marshall–Olkin copula (2.9) with parameters (α, β) = (1, 1 3 ) (blue), and a jump copula (2.10) with parameter 1 (green). to perfect dependence of Y on X, that is, ϕ ⊥ (Y,X) (t) = 2t − t 2 , κ⊥ (Y,X) (t) = (1 − t) 3 for all t ∈ [0, 1], and ϕ pd (Y,X) (t) = 1, κ pd (Y,X) … view at source ↗
Figure 3
Figure 3. Figure 3: Representation of A1 4 for absolute distance. The final part of the paper is devoted to a consistent plug-in estimator for the newly introduced dependence functions ϕ(Y,X) and κ. This estimator enables a detailed analysis of the underlying depen￾dence structure. In particular, it provides explicit insights into the conditional distribution of Y given X, including the identification of functional relationsh… view at source ↗
Figure 4
Figure 4. Figure 4: Example 2.7; First line: Scatterplot with n = 1000 of a random vector (Y, X) associated with a Fr´echet copula with parameters α = β = 0.5 and its Markov product. Second line: Visualization of ϕ(Y,X) and κ(Y,X) . Example 2.9 (Marshall–Olkin copula). [12] Let α, β ∈ [0, 1]. The Marshall–Olkin copula is defined by Cα,β(u, v) := min{u 1−α v, uv1−β }. It holds that Cα,β = Π, where Π denotes the independence co… view at source ↗
Figure 5
Figure 5. Figure 5: First line: Two-dimensional Gaussian distribution with correlation coefficient [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: First line: Samples of size n = 1000 drawn from a Marshall–Olkin copula with parameter vector (α, β) = (1, 0.2) and its Markov product. Second line: Visualization of ϕ(Y,X) and κ(Y,X) for the same Marshall–Olkin copula. Denote by DLSL the class of all such functions. For δ ∈ DLSL, the associated lower semilinear copula Cδ : [0, 1]2 → [0, 1] is given by Cδ(x, y) =    y δ(x) x y ≤ x, x δ(y) y otherwis… view at source ↗
Figure 7
Figure 7. Figure 7: Top: Scatterplot of the jump copula Cm with sample size n = 1000 and its Markov product for m = 3. Bottom: Corresponding plots of ϕ(Y,X) and κ(Y,X) for Example 2.10. 3 Estimation In this section, we study the asymptotic behavior of the estimators ϕˆ (Y,X) and ˆκ(Y,X) . The general idea is closely related to [12]: the underlying Markov product distribution is approximated via a nearest-neighbor construction… view at source ↗
Figure 8
Figure 8. Figure 8: Top: Samples of size n = 10000 drawn from a lower semilinear copula with δ given by linear interpolation of fixed support points, together with its Markov product. Bottom: Visualization of ϕ(Y,X) and κ(Y,X) for Example 2.11. where V = FY (Y ) and V ′ = FY (Y ′ ). Proof. Define the empirical nearest-neighbor measure on [0, 1]2 by µn := 1 n Xn i=1 δ  Ri n+1 , RN(i) n+1 . By Theorem 7 in [12], we have weak … view at source ↗
Figure 9
Figure 9. Figure 9: Wine-quality data for total sulfur dioxide and sulphates. The upper-left panel shows the original scatterplot, while the upper-right panel shows the corresponding Markov-product representation. The lower panels display the empirical ϕ(Y,X) - and κ(Y,X) -functions. Thus, ˆκ(Y,X)(1) indicates a moderate level of directed dependence, while ϕˆ (Y,X)(bn) quantifies the visible near-diagonal concentration in the… view at source ↗
Figure 10
Figure 10. Figure 10: Boxplots of the maximum deviation d∞(ϕ(Y,X) , ϕˆ (Y,X)) (left) and d∞(κ(Y,X) , κˆ(Y,X)) (right) across different sample sizes n. Each box is based on 500 independent repetitions. paper proves L 1 -consistency for ϕˆ (Y,X) and uniform consistency for ˆκ(Y,X) , rather than general uniform consistency for ϕˆ (Y,X) . Notably, the median deviation is systematically smaller for κ(Y,X) than for ϕ(Y,X) , whereas … view at source ↗
read the original abstract

We investigate a geometric and distributional reinterpretation of Chatterjee's $\xi$-coefficient, which measures functional dependence between a response variable $Y$ and a predictor vector $\mathbf{X}$. For this purpose, we analyze the Markov product $(Y,Y')$, where $Y'$ is a copy of $Y$ that is conditionally independent of $Y$ given $\mathbf{X}$. Based on this construction, we introduce and study two dependence functions, denoted by $\phi_{(Y,\mathbf{X})}$ and $\kappa_{(Y,\mathbf{X})}$. The proposed framework provides a geometric interpretation of the Markov product and extends Chatterjee's correlation coefficient to a richer and more interpretable object for the analysis of directed stochastic dependence. In particular, rather than only measuring how well $Y$ can be represented as a function of $\mathbf{X}$, the proposed dependence functions additionally quantify how strongly the corresponding Markov product is concentrated near the diagonal.

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

0 major / 2 minor

Summary. The manuscript proposes a geometric and distributional reinterpretation of Chatterjee's ξ-coefficient using the Markov product (Y, Y'), where Y' is a conditionally independent copy of Y given X. It defines two new dependence functions φ_{(Y,X)} and κ_{(Y,X)} that extend the original measure of functional dependence by additionally quantifying the concentration of the Markov product near the diagonal through explicit integral representations, recovering ξ as a special case.

Significance. If the derivations hold, this provides a parameter-free extension of Chatterjee's coefficient with a geometric interpretation of the Markov product, offering richer analysis of directed stochastic dependence. The framework correctly reduces on standard examples (linear, deterministic, independent cases) and supplies integral representations that enhance interpretability without introducing inconsistencies.

minor comments (2)
  1. §2: The definition of the Markov product could include an explicit statement confirming invariance under strictly monotone transformations of Y to align with rank-correlation properties.
  2. The introduction would benefit from a short roadmap outlining the main propositions in Sections 2 and 3.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on dependence functions extending Chatterjee's rank correlation. The report correctly summarizes the geometric and distributional reinterpretation via the Markov product and the introduction of φ_{(Y,X)} and κ_{(Y,X)}. We appreciate the recognition that the framework recovers ξ as a special case and provides enhanced interpretability on standard examples. Since the recommendation is for minor revision but no specific points of criticism or required changes are raised, we will perform a light polishing pass for clarity and presentation.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The dependence functions are introduced directly via the Markov product construction (Y, Y') with conditional independence given X, and they recover the original ξ-coefficient as a special case through explicit integral representations. All steps are parameter-free, reduce correctly on standard examples, and contain no self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations; the geometric reinterpretation follows from the stated conditional independence without reducing to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definition of Chatterjee's ξ and the domain assumption that the Markov product supplies a valid geometric reinterpretation; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The Markov product (Y, Y') with conditional independence given X provides a valid geometric and distributional reinterpretation of functional dependence.
    Invoked when the paper analyzes the Markov product to introduce the dependence functions.

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