In recent work, Azadkia and Chatterjee 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 (2021). 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,它们完全非对称方法根据等级和最近的相邻图表构成统计数据。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对称一致性,这迅速推动了旨在研究其统计效率的后续工作。在本文件中,我们采用了有条件独立的有条件随机化测试框架,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:对准四边平均可区别替代品和不可对称的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,也依然效率低下,并且作为制定方法变异的动力;参见Lin和Han(2021年)。作为副产品,我们通过证明被视为有条件依赖系数的核心限制词,并附有明确公式来说明非症状差异,从而解决Azadkia和Chatterjee的直方。