AI 中文总结
本文提出Chatterjee秩相关的改进依赖度量，推导其等价表示、构造估计量并证明其强相合性与渐近分布，蒙特卡洛模拟显示该估计量在有限样本性能上优于原度量，I类错误控制更佳。
AI 中文摘要
在近期的突破性研究[JASA, 2021]中，Chatterjee提出了一种基于秩的相关系数ξ(X,Y)，用于衡量随机变量Y对X的依赖关系。与Pearson、Spearman或Kendall等经典度量不同，ξ满足当且仅当X与Y相互独立时ξ=0，当且仅当Y是X的可测函数时ξ=1，无需满足单调性或线性条件。本文提出了ξ(X,Y)的一种改进度量，并研究其理论性质。我们推导了若干等价表示形式，构造了其估计量，证明了该估计量的强相合性及渐近分布，还给出了所提框架的自然扩展形式。蒙特卡洛模拟结果显示，所提估计量在有限样本性能上优于Chatterjee秩相关，尤其在原假设下的I类错误控制方面表现更优。
英文摘要
In his recent breakthrough work [JASA, 2021], Chatterjee proposed a rank-based correlation coefficient $ξ(X,Y)$ to measure the dependence of a random variable $Y$ on $X$. Unlike classical measures such as Pearson, Spearman, or Kendall, $ξ$ satisfies $ξ=0$ if and only if $X$ and $Y$ are independent, and $ξ=1$ if and only if $Y$ is a measurable function of $X$, without requiring monotonicity or linearity. This paper proposes a refined measure of $ξ(X,Y)$ 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.
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