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

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.