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

TL;DR

Chatterjee's correlation coefficient, while useful for measuring association, lacks weak continuity, leading to limitations in statistical inference, especially in tests for independence.

Contribution

This paper demonstrates the lack of weak continuity of Chatterjee's correlation coefficient and discusses its implications for statistical testing.

Findings

Chatterjee's coefficient equals one iff one variable is a measurable function of the other.

The coefficient is not continuous under weak convergence.

Asymptotic tests based on this coefficient can have trivial power against certain alternatives.

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.