On Azadkia–Chatterjee’s conditional dependence coefficient
On Azadkia–Chatterjee’s conditional dependence coefficient
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DOI：
10.3150/22-bej1529
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发表时间：
2021-08
期刊：
影响因子：
1.5
通讯作者：
Hongjian Shi;M. Drton;Fang Han
中科院分区：
文献类型：
--
作者：
Hongjian Shi;M. Drton;Fang Han
在最近的工作中，Azadkia和Chatterjee(2021)提出了一种巧妙的方法来定义条件依赖的一致度量。他们的完全非参数方法形成了基于排名和最近邻图的统计。所得到的条件依赖度量和相关的经验条件依赖系数的吸引人的非参数一致性迅速促进了寻求研究其统计效率的后续工作。本文建立了条件独立性的条件随机化检验(CRT)框架，并对两类局部备选方案，即参数二次均值可微备选方案和非参数H“老光滑备选方案进行了功率分析。我们的局部功率分析表明，即使在CRT框架的辅助下，使用Azadkia-Chatterjee系数的条件独立性检验仍然是低效的，这是发展该方法的变种的动机；参见Lin和han(2022b)。作为副产品，我们通过证明所考虑的条件相关性系数的中心极限定理来解决Azadkia和Chatterjee的猜想，并给出了渐近方差的显式公式。
In recent work, Azadkia and Chatterjee (2021) laid out an ingenious approach to defining consistent measures of conditional dependence. Their fully nonparametric approach forms statistics based on ranks and nearest neighbor graphs. The appealing nonparametric consistency of the resulting conditional dependence measure and the associated empirical conditional dependence coefficient has quickly prompted follow-up work that seeks to study its statistical efficiency. In this paper, we take up the framework of conditional randomization tests (CRT) for conditional independence and conduct a power analysis that considers two types of local alternatives, namely, parametric quadratic mean differentiable alternatives and nonparametric H\"older smooth alternatives. Our local power analysis shows that conditional independence tests using the Azadkia--Chatterjee coefficient remain inefficient even when aided with the CRT framework, and serves as motivation to develop variants of the approach; cf. Lin and Han (2022b). As a byproduct, we resolve a conjecture of Azadkia and Chatterjee by proving central limit theorems for the considered conditional dependence coefficients, with explicit formulas for the asymptotic variances.