A Modified Dependence Measure Related to Chatterjee's Rank Correlation: Theoretical Properties and Asymptotic Analysis

Chuancun Yin

REVIEW 3 major objections 4 minor 40 references

A Modified Dependence Measure Related to Chatterjee's Rank Correlation: Theoretical Properties and Asymptotic Analysis

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proposes a nonnegative modification of the DSS dependence measure and derives an asymptotic chi-square-mixture null limit for its estimator.

desk verdict A well-intentioned reformulation of Chatterjee's correlation whose central null distribution theorem is false. read the letter →

arxiv 2608.07844 v1 pith:U6BBU7RN submitted 2026-08-08 math.ST stat.TH

Chuancun Yin This is my paper · ORCID

classification math.STstat.TH MSC 62G2062H2062G10

keywords dependencemeasureDSSChatterjeerankcorrelationmean-varianceindexstatisticsasymptoticdistributionstrongconsistencyindependencetest

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 proposes a modification of the Dette–Siburg–Stoimenov dependence measure, replacing the left-continuous CDF with a right-continuous one, so the population quantity is well-defined for discrete and mixed distributions. The new coefficient still characterizes independence (value 0) and functional dependence (value 1) exactly, and the paper's plug-in estimator is designed to stay nonnegative, fixing the sign problem of Chatterjee's rank correlation, whose null distribution puts mass on negative values. The paper claims strong consistency for the estimator, an n-rate null limit that is an infinite chi-square mixture, and sqrt-n asymptotic normality under fixed alternatives, plus a family of generalizations. If those claims hold, the estimator gives an independence test with better finite-sample Type I error control than Chatterjee's coefficient.

What carries the argument

The load-bearing object is the ratio $\xi_+(X,Y) = \int \operatorname{Var}(E(1\{Y>t\}|X))\,dF_Y(t) \,/\, \int \operatorname{Var}(1\{Y>t\})\,dF_Y(t)$, written through right-continuous conditional CDFs. Its squared-distance representation, $\xi_+ = \int E[(F_{Y|X}(t)-F_Y(t))^2]\,dF_Y(t) \,/\, \int [F_Y(t)(1-F_Y(t))]\,dF_Y(t)$, connects the measure to the mean-variance index and lets the empirical version inherit U-statistic asymptotics: the categorical plug-in estimator's identity with six times that index is what transfers the chi-square-mixture null limit, and the same representation provides the influence function behind the non-null normal limit.

What would settle it

Compute equation (3.5) when $X$ has a single category ($R=1$) and $Y$ is continuous: the proposed estimator equals $3/n + 1/n^2$, while the mean-variance index is identically $0$; $n$ times the estimator therefore tends to $3$, not to the claimed null limit $0$. This one calculation is enough to test whether the premise of Theorem 3.2 is valid.

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Extended reading notes

Core claim

The central discovery, stated on the paper's own terms, is that replacing the left-continuous CDF in the DSS measure by the right-continuous CDF yields a measure $\xi_+$ that coincides with the original at the extremes—zero exactly under independence and one exactly when $Y$ is a measurable function of $X$—while satisfying cleaner analytic conventions. For a categorical $X$ with $R$ classes, the paper identifies $\xi_+$ with six times the mean-variance index, and uses the known null distribution of that index to claim that under $H_0$, $n\xi_{+,n}$ converges in distribution to $6\sum_{j=1}^\infty \chi^2_j(R-1)/(\pi^2 j^2)$. It further claims strong consistency of the plug-in estimator and asymptotic normality with variance $36\sigma_\tau^2$ under fixed alternatives, and presents a parametrized family $\xi_{k,l,r,s}$ together with a conditional-dependence analogue.

Load-bearing premise

The null-distribution theorem depends on the proposed estimator being exactly six times an earlier mean-variance index; in the simplest one-category case the two are not equal, so the theorem's premise is the equality itself.

Editorial extensions

If this is right

Under $H_0$, for fixed $R$, the test statistic $n\xi_{+,n}$ has a non-normal limit that is an infinite mixture of chi-square variables, so critical values can be computed without bootstrapping.
For alternatives, $\xi_{+,n}$ is consistent and asymptotically normal at $\sqrt{n}$ rate, so confidence intervals for dependence strength follow from a plug-in variance estimator.
Because $\xi_+$ is nonnegative and vanishes only under independence, the proposed test repairs the finite-sample sign problem of Chatterjee's coefficient, which is negative about half the time under the null.
The extension family $\xi_{k,l,r,s}$ preserves the 0/1 characterization and can be tuned through $k,l,r,s$ for different distributional sensitivity.
The conditional version $\xi_+(Y,Z|X)$ avoids left limits and is estimated by plugging out-of-sample conditional CDFs, giving a cleaner asymptotic theory for conditional independence.

Reading between the lines

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

If the stated equality with the mean-variance index fails for degenerate category counts, a centered or debiased version of the estimator would be needed to preserve the chi-square-mixture calibration.
The null limit's dependence on $R$ suggests that permutation or bootstrap critical values could be compared against the mixture quantiles, giving a robustness check that does not rely on the exact equality.
Because $\xi_+$ is an integrated squared distance between conditional and marginal CDFs, the same construction should extend to multivariate $Y$ or vector-valued $X$ with kernel estimators, where the rank-based baseline does not directly apply.
A direct simulation with $R=2$ and continuous $Y$, comparing empirical quantiles of $n\xi_{+,n}$ under $H_0$ to the claimed mixture, would reveal the theorem's practical scope at moderate sample sizes.

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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

Chuancun Yin's paper is a serious but flawed attempt to fix a small defect in Chatterjee's correlation. The central null distribution theorem is false, and the error is not a gap in the proof but an algebraic identity that doesn't hold. I'd send it back for major revision.

What's genuinely good: the paper gives proper credit. It explicitly attributes the population measure to Dette, Siburg, and Stoimenov and notes the equivalent representation from Gamboa, Klein, and Lagnoux. The equivalent representations are clean, and the extension to conditional dependence via plug-in CDF estimates is a reasonable direction. The consistency proof for the kernel estimator (Theorem 3.1) looks plausible under the stated conditions. The simulations are honestly reported with bias and standard deviation tables.

The problem is Theorem 3.2. The estimator (3.5) is not equal to six times the Cui–Zhong tau-hat. The stress-test note gives a concrete counterexample: for R=1, which the theorem allows, (3.5) equals 3/n + 1/n^2, so n times it converges to 3, while the claimed mixture limit with R-1=0 degrees of freedom is 0. The proof's identity would require ∫F_hat^2 dF_hat = 1/3 exactly; for continuous Y this is 1/3 + O(1/n), and the O(1/n) terms survive multiplication by n. So the null distribution is false as stated. That also invalidates the Type I error claims. The simulation section never computes an actual Type I error rate; it only compares variance under the null, which is not the same as calibration.

There are smaller issues. Remark 2.1's claim that xi+ = xi = rho^2 for binary Y is not true in general; it holds only when the conditional mean is linear in X (e.g., binary X). Section 3.5's local power claim uses a normal limit under local alternatives that the paper itself says cannot hold under H0; the argument is hand-wavy.

Net: the paper is a sincere contribution with careful attributions and a plausible extension, but the main asymptotic result is wrong. A referee should be asked to verify the identity, because if I'm right, the test is not calibrated and the paper needs major revision. This is worth refereeing, not desk-rejecting, because the error is subtle and the rest of the paper is competent.

Referee Report

3 major / 4 minor

Summary. The paper proposes a modified DSS-type dependence measure ξ+(X,Y), gives equivalent representations, constructs sample estimators for categorical and continuous X, and claims strong consistency, an asymptotic null distribution as a scaled infinite chi-square mixture for categorical X, asymptotic normality under fixed alternatives, and an extension to conditional dependence. The central theoretical result is Theorem 3.2, which states that under H0: X⊥Y, nξ+,n converges to 6Σ_j χ²_j(R-1)/(π²j²) for the estimator in (3.5). The paper also reports simulations suggesting improved Type I error control relative to Chatterjee's coefficient.

Significance. If Theorem 3.2 were correct, the paper would offer a nonnegative estimator of a dependence measure with a known null distribution for categorical X, thereby addressing a known drawback of Chatterjee's coefficient. The paper properly credits earlier work (Gamboa, Klein and Lagnoux; Cui and Zhong) rather than creating a citation loop. However, the main asymptotic claim is false: the identity on which the proof rests is algebraically incorrect, so the proposed test is not asymptotically calibrated. The claimed theoretical contribution is therefore not established, and the favorable Type I error statements in the abstract and conclusions are not supported.

major comments (3)

[Section 3.2, Theorem 3.1(ii)] The proof of strong consistency for continuous X is not a valid proof. It asserts 'by the strong law of large numbers' that R_i/n → F(Y_i) and R_{i,j}/n_j → F(Y_i|X_j), but these are not sums of i.i.d. random variables for fixed indices i,j; the observations Y_i and X_j depend on n, the kernel weights involve the bandwidth h_n, and the denominator n_j can be zero or small. The regularity conditions (C1)–(C4) are not used anywhere in the proof. Consequently, the almost-sure limit for the kernel estimator is unproven, which also undermines Theorem 3.3 and the simulation-based claims for the kernel version of the estimator.
[Section 3.5, Asymptotic Power] The local alternative claim is unsupported. The paper simply states that under H1n: ξ+ = δ/√n, √nξ+,n → N(δ, σ²), without proving contiguity, deriving the limiting variance under the local sequence, or verifying that the fixed-alternative variance σ² applies. Moreover, the proposed test statistic uses the standardization √nξ+,n/σ̂_n and normal quantiles, whereas under H0 the correct scaling is n (as claimed in Theorem 3.2). Because Theorem 3.2 is false, the description of the level-α test and the power function β(δ) is not justified.
[Section 6, Table 2 and the Abstract] The simulation study uses the kernel estimator (3.2) with continuous X, but the only null distribution provided in the paper, Theorem 3.2, applies to the categorical estimator (3.5). No null limit is derived for the kernel estimator, so the claimed 'better Type I error control' (Abstract and Section 7) is not supported by any theoretical result. The r=0 rows of Table 2 report means of ξ+,n, yet these are not compared with the claimed mixture distribution or with the correct null limit, so the simulation does not validate the asymptotic calibration of the test.

minor comments (4)

[Section 2, Proposition 2.2 proof] In the proof of necessity of (iii), the displayed expression '∫ E[(F^2_{Y|X}(t) − F_Y(t)]dF_Y(t) = 0' is a typo; it should be '∫ (E[F_{Y|X}(t)^2] − F_Y(t)^2) dF_Y(t) = 0' (or the equivalent form derived from (2.5)). The subsequent reasoning is understandable but the formula as written is not.
[Throughout] There are numerous typos and grammatical errors, including 'remak' (Introduction), 'As showed by Chatterjee' (Introduction), 'as follow' (Introduction), and 'the sample estimator of ξ+(X,Y) defined in (2.2)' (Section 3.1), where (2.2) is a population expression. A careful proofreading pass is needed.
[Sections 3.1–3.5] The same symbol ξ+,n is used for the categorical estimator (3.1)/(3.5), the kernel estimator (3.2), and the conditional-dependence estimator (5.1), without explicit distinction. This creates ambiguity in Theorems 3.2–3.4 and in the simulation section, where the reader must infer which estimator is being discussed.
[Section 3.4, Theorem 3.4] The statement 'Since ξ+,n = 6τ̂n + o_p(n^{-1/2}) by construction' is not a consequence of the construction; the exact relation is ξ+,n = 6τ̂n + 3/n + 1/n², which is indeed o_p(n^{-1/2}) under H1. The proof should be corrected to state and use the exact relation, rather than an unsupported identity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Theorem 3.2 issue is a non-circular algebraic mismatch with an external theorem, not a derivation that reduces to its own inputs.

full rationale

The paper's derivation chain does not exhibit circularity. The proposed measure ξ+ is defined directly in Definition 2.1, and the population identity ξ+ = 6τ in equation (3.4) is an algebraic expansion of the mean-variance index of Cui and Zhong, an external result cited from [13]. The estimator in (3.5) is a plug-in estimator, and the proof of Theorem 3.2 attempts to apply Cui-Zhong's Theorem 1 to it. Whether the plug-in estimator actually equals 6τ̂ is a factual algebraic question; as the reader notes, the equality fails because ∫F̂²dF̂ = 1/3 + O(1/n) rather than exactly 1/3, so nξ_{+,n} need not converge to the stated limit for the estimator in (3.5). This is a non-circular mathematical error, or a mismatch between estimators, not a case of a fitted parameter renamed as a prediction, a self-citation chain, or a definition that smuggles in the conclusion. The paper's key external inputs—Cui and Zhong's limit theorem, Chatterjee's consistency result, Gamboa et al.'s representation—are independent of the present paper's own claims, and no load-bearing step reduces to the paper's own output. The paper also contains no relevant self-citations. Therefore the appropriate circularity score is 0; the correctness defect in Theorem 3.2 is a soundness issue, not a circularity issue.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central asymptotic claims rest entirely on external theorems (Cui-Zhong) and on an unproved uniform-convergence step for kernel conditional CDF estimators. The proposed measure itself is a reformulation of existing quantities, and the main theorem is misstated for the estimator it names.

free parameters (1)

Kernel bandwidth h = 1.06 sigma_hat_X n^{-1/5}
Bandwidth for kernel conditional CDF estimation in the continuous-X estimator. Chosen by a rule of thumb, not fitted to data, but the estimator's finite-sample behavior and the proof of consistency depend on it.

assumptions (3)

standard math Variance decomposition Var(A) = Var(E(A|X)) + E(Var(A|X)) for indicator variables.
Used throughout Proposition 2.1 to derive equivalent representations of xi_plus.
domain assumption Cui and Zhong's Theorems 1 and 2 give the null and alternative asymptotic distributions of the mean-variance index tau_hat.
The paper's central asymptotic results are direct citations of these theorems; if they are misstated or inapplicable to the actual estimator, the asymptotic claims fail.
domain assumption Under conditions C1-C4, kernel conditional CDF estimators converge uniformly, justifying the SLLN step for the double sum in Theorem 3.1(ii).
The proof of Theorem 3.1(ii) asserts termwise convergence and does not prove the uniform convergence needed to pass from pointwise limits to the integral.

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Pith. "Pith review of A Modified Dependence Measure Related to Chatterjee's Rank Correlation: Theoretical Properties and Asymptotic Analysis." pith.science (2026). https://pith.science/paper/U6BBU7RN

@misc{pith2026260807844,
  author       = {Pith},
  title        = {Pith review of: A Modified Dependence Measure Related to Chatterjee's Rank Correlation: Theoretical Properties and Asymptotic Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U6BBU7RN}},
  note         = {Machine review of arXiv:2608.07844}
}

abstract

In his recent breakthrough work [JASA, 2021], Chatterjee proposed a rank-based correlation coefficient $\xi(X,Y)$ to measure the dependence of a random variable $Y$ on $X$. Unlike classical measures such as Pearson, Spearman, or Kendall, $\xi$ satisfies $\xi=0$ if and only if $X$ and $Y$ are independent, and $\xi=1$ if and only if $Y$ is a measurable function of $X$, without requiring monotonicity or linearity. This paper proposes a refined measure of $\xi(X,Y)$ and investigates its theoretical properties. We derive several equivalent representations, construct an estimator, and establish its strong consistency as well as its asymptotic distribution. A natural extension of the proposed framework is also presented. The Monte Carlo simulations show that the proposed estimator outperforms Chatterjee's rank correlation in terms of finite-sample performance, particularly in controlling Type I error under the null.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Auddy, A., Deb, N., & Nandy, S. (2024). Exact detection thresholds and minimax optimality of Chatterjee’s correlation. Bernoulli 30(2), 1056-1079

work page 2024

[2] [2]

Ansari, J., Langthaler, Patrick B., Fuchs, S., & Trutschnig, W. (2026). Quantifying and estimating dependence via sensitivity of conditional distributions. Bernoulli 32 (1): 179-204

work page 2026

[3] [3]

Ansari, J., & Rockel, M. (2026). The exact region and an inequality between Chatter- jee’s and Spearman’s rank correlations. Journal of Multivariate Analysis 214, 105630

work page 2026

[4] [4]

Azadkia, M., & Chatterjee, S. (2021). A simple measure of conditional dependence. Annals of Statistics 49(6), 3070-3102

work page 2021

[5] [5]

Bahari, F. (2026). Improvement of Chatterjee’s correlation with missing at random data in theYvariable. Statistical Papers (2026) 67: 56

work page 2026

[6] [6]

Bickel, P. J. (2022). Measures of independence and functional dependence. Available at arXiv:2206.13663

arXiv 2022

[7] [7]

Cao, S., & Bickel, P. J. (2020). Correlations with tailored extremal properties. Avail- able at arXiv:2008.10177v2

arXiv 2020

[8] [8]

& Liang, W.Q

Chao, C.C., Bai, Z. & Liang, W.Q. (1993). Asymptotic normality for oscillation of permutation. Probability in the Engineering and Informational Sciences 7, 227-235

work page 1993