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On the lack of weak continuity of Chatterjee's correlation coefficient

Axel Bücher, Holger Dette

Abstract

Chatterjee's correlation coefficient has recently been proposed as a new association measure for bivariate random vectors that satisfies a number of desirable properties. Among these properties is the feature that the coefficient equals one if and only if one of the variables is a measurable function of the other. As already observed in Mikusinski, Sherwood and Taylor (Stochastica, 13(1):61-74, 1992), this property implies that Chatterjee's coefficient is not continuous with respect to weak convergence. We discuss a number of negative consequences for statistical inference. In particular, we show that asymptotic tests for stochastic independence based on Chatterjee's empirical correlation coefficient, or boosted versions thereof, have trivial power against certain alternatives for which the population coefficient is one.

On the lack of weak continuity of Chatterjee's correlation coefficient

Abstract

Chatterjee's correlation coefficient has recently been proposed as a new association measure for bivariate random vectors that satisfies a number of desirable properties. Among these properties is the feature that the coefficient equals one if and only if one of the variables is a measurable function of the other. As already observed in Mikusinski, Sherwood and Taylor (Stochastica, 13(1):61-74, 1992), this property implies that Chatterjee's coefficient is not continuous with respect to weak convergence. We discuss a number of negative consequences for statistical inference. In particular, we show that asymptotic tests for stochastic independence based on Chatterjee's empirical correlation coefficient, or boosted versions thereof, have trivial power against certain alternatives for which the population coefficient is one.

Paper Structure

This paper contains 4 sections, 6 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Shuffles of the Min and measures of perfect dependence
  3. Consequences for testing independence
  4. Consequences for confidence intervals

Key Result

Corollary 1.1

For any pair of independent random variables $(X, Y)$ with continuous marginal cdfs and any $\xi_0 \in [0,1]$, there exists a sequence of random variables $(X_k, Y_k)$ weakly converging to $(X,Y)$ for $k\to\infty$ such that Chatterjee's correlation coefficient satisfies $\xi(X_k, Y_k)=\xi_0$ for all

Theorems & Definitions (12)

  • Corollary 1.1
  • Theorem 2.1: KimSam78MikShe92
  • Corollary 2.2
  • Proposition 2.3
  • proof : Proof of Proposition \ref{['prop:Xi']}
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • ...and 2 more