Abstract：The Azadkia-Chatterjee coefficient is a rank-based measure of dependence between a random variable and a random vector. In this paper, we extend it to a measure of the dependence between two random vectors Y and Z based on n i.i.d. samples. The proposed coefficient converges almost surely to a limit with the following properties: i) it lies in [0, 1]; ii) it is equal to zero if and only if Y and Z are independent; and iii) it is equal to one if and only if Y is almost surely a function of Z. Remarkably, the only assumption required by this convergence is that Y is not almost surely a constant vector. We further prove that under the same mild condition and after a proper scaling, this coefficient converges in distribution to a standard normal random variable when Y and Z are independent. This asymptotic normality result allows us to construct a Wald-type hypothesis test of independence based on this coefficient. To compute this coefficient, we propose a merge sort based algorithm that runs in O(n (log n)^{dim(Y)}). Finally, we show that it can be used to measure the conditional dependence between Y and Z conditional on a third random vector X, and prove that the measure is monotonic with respect to the deviation from an independence distribution under certain model restrictions.

Bio：王禹皓，清华大学交叉信息学院助理教授。本科毕业于清华大学自动化系，随后进入麻省理工学院计算机和电子工程系攻读博士学位，并任职于LIDS实验室。王禹皓教授在入职清华大学之前任职于剑桥大学统计学实验室并担任博士后研究员。王禹皓教授目前的研究兴趣集中在：因果推断、实验设计、高维统计、免分布假设检验等领域。王禹皓教授曾有多篇文章发表于The Annals of Statistics，JRSSB，Biometrika，JOE，Bernoulli等顶尖统计与计量经济学学期刊以及NeurIPS等顶尖机器学习与人工智能会议。王禹皓现担任统计学期刊Electronic Journal of Statistics的副编辑职务，并曾入选福布斯中国2021年度30 Under 30榜单：科学和医疗健康榜单。