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arxiv: 2406.16859 · v2 · pith:A7F2XIEUnew · submitted 2024-06-24 · 📊 stat.ME

On the extensions of the Chatterjee-Spearman test

Qingyang Zhang This is my paper

Pith reviewed 2026-05-23 23:34 UTC · model grok-4.3

classification 📊 stat.ME
keywords independence testingrank correlationsChatterjee correlationSpearman correlationKendall tauasymptotic normalitysymmetrized testmultivariate extension
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The pith

The symmetrized Chatterjee-Spearman test has a derived asymptotic null distribution, and Chatterjee's correlation is asymptotically independent of Kendall's tau under independence.

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

The paper extends a max-type independence test that combines Chatterjee's correlation with Spearman's to improve power against monotonic alternatives. It first symmetrizes the test statistic and derives its limiting null distribution. It then proves that Chatterjee's correlation is asymptotically jointly normal and independent of both Kendall's tau and quadrant correlation when variables are independent. Simulations show the Chatterjee-Kendall version detects dependence more effectively than the Spearman version. The work also sketches two routes to multivariate versions of the combined test.

Core claim

Under the null of independence the symmetrized max-type statistic converges to a known limiting distribution; Chatterjee's correlation is asymptotically independent of Kendall's tau and of quadrant correlation; the Chatterjee-Kendall combination attains higher power than the Chatterjee-Spearman combination in finite samples; and the same joint-normality framework supports two distinct multivariate extensions.

What carries the argument

The symmetrized max-type statistic formed from Chatterjee's and Spearman's rank correlations, together with the asymptotic joint normality result under independence.

If this is right

  • Critical values for the symmetrized test can be obtained from the derived limiting distribution without further simulation.
  • The Chatterjee-Kendall combination can replace the original test when higher power against monotonic dependence is desired.
  • The asymptotic independence result extends immediately to any other rank correlation that satisfies the same regularity conditions.
  • The two sketched multivariate extensions make the combined test usable for vector-valued observations.

Where Pith is reading between the lines

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

  • The independence between Chatterjee's correlation and classical rank measures suggests the former captures non-monotonic or non-linear features that the latter miss.
  • One could form a three-way max-type statistic using Chatterjee, Kendall, and quadrant correlations without inflating the null variance.
  • The multivariate extensions could be checked on real data sets with known dependence structure to see whether power gains persist in dimensions greater than two.

Load-bearing premise

The rank correlations obey the regularity conditions that justify the asymptotic joint normality statements under the null of independence.

What would settle it

A data set or explicit counter-example in which the sample Chatterjee correlation and Kendall tau remain correlated at rate 1/sqrt(n) even after the variables are made independent.

Figures

Figures reproduced from arXiv: 2406.16859 by Qingyang Zhang.

Figure 1
Figure 1. Figure 1: Scatterplots of τn(X,Y) and ξn(X,Y) under n = 30,100,300,500. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Scatterplots of Qn(X,Y) and ξn(X,Y) under n = 30,100,300,500. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_2.png] view at source ↗
read the original abstract

Chatterjee (2021) introduced a novel independence test that is rank-based, asymptotically normal and consistent against all alternatives. One limitation of Chatterjee's test is its low statistical power for detecting monotonic relationships. To address this limitation, in our previous work (Zhang, 2024, Commun. Stat. - Theory Methods), we proposed to combine Chatterjee's and Spearman's correlations into a max-type test and established the asymptotic joint normality. This work examines three key extensions of the combined test. First, motivated by its original asymmetric form, we extend the Chatterjee-Spearman test to a symmetric version, and derive the asymptotic null distribution of the symmetrized statistic. Second, we investigate the relationships between Chatterjee's correlation and other popular rank correlations, including Kendall's tau and quadrant correlation. We demonstrate that, under independence, Chatterjee's correlation and any of these rank correlations are asymptotically joint normal and independent. Simulation studies demonstrate that the Chatterjee-Kendall test has better power than the Chatterjee-Spearman test. Finally, we explore two possible extensions to the multivariate case. These extensions expand the applicability of the rank-based combined tests to a broader range of scenarios.

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

2 major / 0 minor

Summary. The paper extends the Chatterjee-Spearman independence test in three directions: (i) symmetrizing the test and deriving its asymptotic null distribution, (ii) establishing that Chatterjee's rank correlation is asymptotically jointly normal and independent of Kendall's tau and of quadrant correlation under the null of independence, with simulations indicating higher power for the Chatterjee-Kendall combination, and (iii) exploring two multivariate extensions.

Significance. If the asymptotic derivations and simulation comparisons hold, the work would usefully enlarge the toolkit of rank-based independence tests by improving power against monotonic alternatives and by clarifying independence from other classical rank correlations, while also moving toward multivariate settings.

major comments (2)
  1. [Abstract] Abstract: the claim that the symmetrized statistic possesses a derived asymptotic null distribution is asserted without any displayed equations, proof outline, or regularity conditions, so the validity of this central derivation for the first extension cannot be assessed.
  2. [Abstract] Abstract: the assertion that Chatterjee's correlation is asymptotically jointly normal and independent of Kendall's tau (and of quadrant correlation) under independence invokes regularity conditions whose satisfaction for these particular rank statistics is not exhibited or verified; this is load-bearing for the second extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on the abstract. We respond to each major comment below. The technical details are developed in the body of the paper, but we agree that the abstract can be improved for clarity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the symmetrized statistic possesses a derived asymptotic null distribution is asserted without any displayed equations, proof outline, or regularity conditions, so the validity of this central derivation for the first extension cannot be assessed.

    Authors: We agree that the abstract, being a concise summary, does not display the equations or proof outline. The derivation of the asymptotic null distribution of the symmetrized Chatterjee-Spearman statistic (including the explicit limiting variance obtained via the delta method applied to the joint asymptotic normality of the two components, and the verification of the required moment conditions) appears in Theorem 2 and its proof in Section 3. To allow readers to assess the claim from the abstract itself, we will revise the abstract to include a one-sentence outline of the derivation approach and a reference to the theorem. revision: yes

  2. Referee: [Abstract] Abstract: the assertion that Chatterjee's correlation is asymptotically jointly normal and independent of Kendall's tau (and of quadrant correlation) under independence invokes regularity conditions whose satisfaction for these particular rank statistics is not exhibited or verified; this is load-bearing for the second extension.

    Authors: We acknowledge that the abstract does not exhibit the verification. The proof that the relevant regularity conditions (finite fourth moments of the kernel functions and the resulting zero asymptotic covariance under independence) hold for Chatterjee's statistic together with Kendall's tau and the quadrant correlation is given explicitly in Section 4, with the covariance calculations shown in the appendix. We will revise the abstract to state that the conditions are verified for these specific rank correlations and to cite the relevant result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new extensions presented as independent demonstrations

full rationale

The abstract cites prior self-work (Zhang 2024) only for the base Chatterjee-Spearman joint normality result. The three extensions—symmetrized statistic null distribution, joint normality/independence with Kendall/quadrant correlations, and multivariate cases—are explicitly described as new demonstrations and simulation studies performed in this paper. No equations or proof reductions are supplied that would allow any new claim to be shown as equivalent to the cited prior result by construction. Self-citation is present but does not carry the load-bearing argument for the novel contributions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, background axioms, or new postulated entities; all such details are absent from the available text.

pith-pipeline@v0.9.0 · 5704 in / 1138 out tokens · 51704 ms · 2026-05-23T23:34:48.567351+00:00 · methodology

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Reference graph

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