Kernel Estimation Of Chatterjee's Dependence Coefficient

Abstract

A dependence coefficient suggested by Chatterjee (2021) has attracted considerable attention in recent years. It takes values between 0 and 1, vanishes if and only if the corresponding random variables $X$ and $Y$ are independent, and equals 1 if and only if $Y$ is almost surely a measurable function of $X$. The coefficient coincides, for distributions with continuous marginals, with a dependence coefficient introduced in Dette, Siburg, and Stoimenov (2013). Several estimators of this measure have been proposed, including the kernel-type estimator of Dette, Siburg, and Stoimenov (2013), Chatterjee's rank correlation, and its nearest-neighbour modification by Lin and Han (2023). In this paper, we investigate the asymptotic properties and local power of the kernel-type estimator. We first establish a non-degenerate central limit theorem under independence: after suitable centring, the estimator is asymptotically normal at the rate $\sqrt{n/h_1}$, where $h_1\to 0$ is a bandwidth. We then consider Gaussian-copula alternatives with correlation parameter ${\rho }_n\to 0$. By deriving a second-order expansion of the mean, establishing variance stability and proving a central limit theorem under local alternatives, we obtain an explicit limiting power function and identify $(h_1/n{)}^{1/4}$ as the detection boundary. Under suitable bandwidth conditions, the detection boundary can attain rates arbitrarily close to $n^{-3/8}$. The exponent $3/8$ exceeds the exponent $1/4$ for the test based on Chatterjee's original rank correlation and the exponent approaching $5/16$ for the modified rank correlation of Lin and Han (2023) within the range of their central limit theorem. The proofs combine refined results for double-indexed permutation statistics with Gaussian rank-score expansions and Hermite-chaos projections. A simulation study illustrates the normal approximation and the resulting power properties.

arXiv commentRevised version with new results on local asymptotic power under Gaussian copula alternatives