University of California, Berkeley（加州大学伯克利分校）；University of Washington（华盛顿大学）
Zhexiao Lin, Fang Han
备注：
Multiple minor improvements were made in this version, including (1) a proof of the existence of the limiting variance, (2) some numeric studies, and (3) an analysis of the Sobol' indices
Establishing the limiting distribution of Chatterjee's rank correlation for a general, possibly non-independent, pair of random variables has been eagerly awaited by many. This paper shows that (a) Chatterjee's rank correlation is asymptotically normal as long as one variable is not a measurable function of the other, (b) the corresponding asymptotic variance is uniformly bounded by 36, and (c) a consistent variance estimator exists. Similar results also hold for Azadkia-Chatterjee's graph-based correlation coefficient, a multivariate analogue of Chatterjee's original proposal. The proof is given by appealing to Hájek representation and Chatterjee's nearest-neighbor CLT.
Chatterjee (2021)'s ingenious approach to estimating a measure of dependence first proposed by Dette et al. (2013) based on simple rank statistics has quickly caught attention. This measure of dependence has the unusual property of being between 0 and 1, and being 0 or 1 if and only if the corresponding pair of random variables is independent or one is a measurable function of the other almost surely. However, more recent studies (Cao and Bickel, 2020; Shi et al., 2021b) showed that independence tests based on Chatterjee's rank correlation are unfortunately rate-inefficient against various loc…
Chatterjee (2021) introduced a simple new rank correlation coefficient that has attracted much recent attention. The coefficient has the unusual appeal that it not only estimates a population quantity first proposed by Dette et al. (2013) that is zero if and only if the underlying pair of random variables is independent, but also is asymptotically normal under independence. This paper compares Chatterjee's new correlation coefficient to three established rank correlations that also facilitate consistent tests of independence, namely, Hoeffding's $D$, Blum-Kiefer-Rosenblatt's $R$, and Bergsma-D…
Limit theorems of Azadkia-Chatterjee's conditional graph correlation
School of Statistics, University of International Business and Economics（对外经济贸易大学统计学院）；Department of Statistics, University of Washington（华盛顿大学统计学系）；Academy of Mathematics and System Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
In this paper we propose and study a class of simple, nonparametric, yet interpretable measures of association between two random variables $X$ and $Y$ taking values in general topological spaces. These nonparametric measures -- defined using the theory of reproducing kernel Hilbert spaces -- capture the strength of dependence between $X$ and $Y$ and have the property that they are 0 if and only if the variables are independent and 1 if and only if one variable is a measurable function of the other. Further, these population measures can be consistently estimated using the general framework of…
This paper is concerned with the limiting spectral behaviors of large dimensional Kendall's rank correlation matrices generated by samples with independent and continuous components. We do not require the components to be identically distributed, and do not need any moment conditions, which is very different from the assumptions imposed in the literature of random matrix theory. The statistical setting in this paper covers a wide range of highly skewed and heavy-tailed distributions. We establish the central limit theorem (CLT) for the linear spectral statistics of the Kendall's rank correlati…
