Azadkia and Chatterjee (2021) recently introduced a simple nearest neighbor (NN) graph-based correlation coefficient that consistently detects both independence and functional dependence. Specifically, it approximates a measure of dependence that equals 0 if and only if the variables are independent, and 1 if and only if they are functionally dependent. However, this NN estimator includes a bias term that may vanish at a rate slower than root-$n$, preventing root-$n$ consistency in general. In this article, we (i) analyze this bias term closely and show that it could become asymptotically negligible when the dimension is smaller than four; and (ii) propose a bias-correction procedure for more general settings. In both regimes, we obtain estimators (either the original or the bias-corrected version) that are root-$n$ consistent and asymptotically normal.

翻译：Azadkia和Chatterjee（2021）近期提出了一种基于最近邻（NN）图的简单相关系数，该系数能够一致地检测独立性与函数依赖性。具体而言，它近似于一种依赖度量：当且仅当变量独立时该度量为0，当且仅当变量存在函数依赖时该度量为1。然而，该NN估计量包含一个偏差项，其收敛速度可能慢于根号n，导致一般情形下无法达到根号n一致性。本文中，我们（i）深入分析该偏差项，证明当维度小于四时该偏差项可能渐近可忽略；（ii）针对更一般场景提出偏差校正方法。在这两种情况下，我们获得的估计量（原始版本或校正版本）均具有根号n一致性与渐近正态性。