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On continuity of Chatterjee's rank correlation and related dependence measures

Jonathan Ansari, Sebastian Fuchs

TL;DR

The paper addresses the lack of weak continuity of Chatterjee's rank correlation $\xi$ by reframing the problem through the Markov product $(Y,Y')$ with $Y'$ conditionally independent of $Y$ given $\mathbf{X}$. It develops a general continuity toolkit based on conditional weak convergence, noise robustness, and copula-based $\partial_1$-convergence to guarantee weak convergence of Markov products and hence continuity of $\xi$ under broad modeling relaxations. It also extends the framework to related dependence measures, including a multivariate extension $T$ and an explainability measure $\Lambda$, proving their continuity in important distributional families such as elliptical and $\ell_1$-norm symmetric distributions and under common copula models. The results yield stability insights for inference under model misspecification and enable continuity guarantees for a range of dependence measures in practical applications. Overall, the work provides a unified, copula- and conditional-distribution-based approach to ensuring robustness of $\xi$-type measures across diverse statistical models.

Abstract

While measures of concordance -- such as Spearman's rho, Kendall's tau, and Blomqvist's beta -- are continuous with respect to weak convergence, Chatterjee's rank correlation xi recently introduced in Azadkia and Chatterjee [5] does not share this property, causing drawbacks in statistical inference as pointed out in Bücher and Dette [7]. As we study in this paper, xi is instead weakly continuous with respect to conditionally independent copies -- the Markov products. To establish weak continuity of Markov products, we provide several sufficient conditions, including copula-based criteria and conditions relying on the concept of conditional weak convergence in Sweeting [36]. As a consequence, we also obtain continuity results for xi and related dependence measures and verify their continuity in the parameters of standard models such as multivariate elliptical and l1-norm symmetric distributions.

On continuity of Chatterjee's rank correlation and related dependence measures

TL;DR

The paper addresses the lack of weak continuity of Chatterjee's rank correlation by reframing the problem through the Markov product with conditionally independent of given . It develops a general continuity toolkit based on conditional weak convergence, noise robustness, and copula-based -convergence to guarantee weak convergence of Markov products and hence continuity of under broad modeling relaxations. It also extends the framework to related dependence measures, including a multivariate extension and an explainability measure , proving their continuity in important distributional families such as elliptical and -norm symmetric distributions and under common copula models. The results yield stability insights for inference under model misspecification and enable continuity guarantees for a range of dependence measures in practical applications. Overall, the work provides a unified, copula- and conditional-distribution-based approach to ensuring robustness of -type measures across diverse statistical models.

Abstract

While measures of concordance -- such as Spearman's rho, Kendall's tau, and Blomqvist's beta -- are continuous with respect to weak convergence, Chatterjee's rank correlation xi recently introduced in Azadkia and Chatterjee [5] does not share this property, causing drawbacks in statistical inference as pointed out in Bücher and Dette [7]. As we study in this paper, xi is instead weakly continuous with respect to conditionally independent copies -- the Markov products. To establish weak continuity of Markov products, we provide several sufficient conditions, including copula-based criteria and conditions relying on the concept of conditional weak convergence in Sweeting [36]. As a consequence, we also obtain continuity results for xi and related dependence measures and verify their continuity in the parameters of standard models such as multivariate elliptical and l1-norm symmetric distributions.

Paper Structure

This paper contains 13 sections, 15 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Continuity of $\xi$
  3. Sufficient conditions for weak convergence of Markov products
  4. Conditional weak convergence
  5. Noise resistance
  6. $\partial_1$-convergence of copulas
  7. Continuity of dependence measures related to $\xi$
  8. An extension of $\xi$ to multivariate output variables
  9. A measure of explainability
  10. Related dependence measures
  11. Continuity results for families of distributions
  12. Continuity results within the class of elliptical distributions
  13. Continuity results within the class of $\ell_1$-norm symmetric distributions

Key Result

Lemma 2.1

For $(Y,Y')$ in Assumption.DimR, Chatterjee's rank correlation satisfies for positive constants $a := (\int_{\mathbb{R}} \mathrm{Var}(\mathds{1}_{\{Y\geq y\}})\mathrm{\,d} P^Y(y))^{-1}$ and $b := a \int_{\mathbb{R}} P(Y<y)^2 \mathrm{\,d} P^Y(y)\,,$ both depending only on $\overline{\mathsf{Ran}(F_Y)}\,.$

Figures (2)

  • Figure 1: Plots of $\xi(Y,\mathbf{X})$ in dependence on the correlation $\rho$ in the equicorrelated normal setting of Example \ref{['exeqnor']} for various dimensions $p\in \{1,2,4,10,100\}$ of $\mathbf{X}\,.$
  • Figure 2: Plots of $T(\mathbf{Y},\mathbf{X})$ for $(\mathbf{X},\mathbf{Y})=(X_1,X_2,Y_1,Y_2)$ being normally distributed (solid lines) and $t$-distributed with $3$ degrees of freedom (dotted) with covariance matrix $\Sigma$ given by \ref{['eq4dnormal']} in dependence on $\rho_{XY}$ for fixed $\rho_X=0.5$ and for several $\rho_{Y}\in \{-0.999,-0.99,-0.9,-0.75,-0.5,0,0.5,0.9\}\,.$

Theorems & Definitions (24)

  • Example 1.1: $\xi$ is not continuous w.r.t. weak convergence of $(X_n,Y_n)$ to $(X,Y)$
  • Lemma 2.1: Representation of $\xi$
  • Theorem 2.2: Continuity of $\xi$ based on Markov products
  • Remark 2.3
  • Example 2.4: $(X_n,Y_n)\xrightarrow{~d~}(X,Y)$ does not imply $(Y_n,Y_n')\xrightarrow{~d~} (Y,Y')$
  • Example 2.5: $(Y_n,Y_n')\xrightarrow{~d~} (Y,Y')$ does not imply $(X_n,Y_n)\xrightarrow{~d~}(X,Y)$
  • Theorem 3.1: Conditional weak convergence
  • Remark 3.2
  • Proposition 3.3: Additive error model
  • Proposition 3.4: Robustness of $\xi$ against small pertubations of the response
  • ...and 14 more