  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.
