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.

翻译：在最近的工作中,Azadkia 和 Chatterjee (2021年) 提出了界定一致的有条件依赖措施的巧妙方法,它们完全非参数方法以等级和最近的相邻图表为基础,构成统计数据。由此产生的有条件依赖措施和相关的经验性有条件依赖系数具有吸引力的非参数一致性,这迅速推动了旨在研究其统计效率的后续工作。在本文中,我们采用了有条件独立有条件随机化测试框架(CRT),并进行了一种权势分析,考虑了两种类型的当地替代方法,即:参数等离差平均可变替代方法和非参数 H\'older 光滑替代方法。我们的地方权力分析表明,即使利用Azadkia-Chatterjee 系数进行有条件独立测试,即使得到CRT框架的帮助,也是效率低下的,而且作为开发方法变式的动力;参见Lin 和 Han (2022b) 。作为副产品,我们通过证明被认为的有条件依赖系数的核心限制,用明确的公式来解决Azadkia和Chatterjee的预测。