Kernel Estimation Of Chatterjee's Dependence Coefficient

Abstract: Dette, Siburg, and Stoimenov (2013) introduced a copula-based measure of dependence, which implies independence if it vanishes and is equal to 1 if one variable is a measurable function of the other. For continuous distributions, the dependence measure also appears as stochastic limit of Chatterjee's rank correlation (Chatterjee, 2021). They proved asymptotic normality of a corresponding kernel estimator with a parametric rate of convergence. In recent work Shi, Drton, and Han (2022) revealed empirically and theoretically that under independence the asymptotic variance degenerates. In this note, we derive the correct asymptotic distribution of the kernel estimator under the null hypothesis of independence. We show that after a suitable centering and rescaling at a rate larger than $\sqrt{n}$ (where $n$ is the sample size), the estimator is asymptotically normal. The analysis relies on a refined central limit theorem for double-indexed linear permutation statistics and accounts for boundary effects that are asymptotically non-negligible. As a consequence, we obtain a valid basis for independence testing without relying on permutations and argue that tests based on the kernel estimator detect local alternatives converging to the null at a faster rate than those detectable by Chatterjee's rank correlation.