May 2024 On Azadkia–Chatterjee’s conditional dependence coefficient
Hongjian Shi, Mathias Drton, Fang Han
Author Affiliations +
Bernoulli 30(2): 851-877 (May 2024). DOI: 10.3150/22-BEJ1529


In recent work, Azadkia and Chatterjee (Ann. Statist. 49 (2021) 3070–3102) 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 Hölder 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 (Biometrika 110 (2023) 283–299). 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.

Funding Statement

The authors have received funding from the United States NSF Grants DMS-1712536 and SES-2019363 and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 883818).


The authors would like to thank two anonymous referees, an anonymous Associate Editor, and the Editor Mark Podolskij for their stimulating comments, which highly improved the quality of this paper.


Download Citation

Hongjian Shi. Mathias Drton. Fang Han. "On Azadkia–Chatterjee’s conditional dependence coefficient." Bernoulli 30 (2) 851 - 877, May 2024.


Received: 1 September 2021; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699537
Digital Object Identifier: 10.3150/22-BEJ1529

Keywords: Conditional independence , Graph-based test , local power analysis , Nearest neighbor graphs , rank-based test


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Vol.30 • No. 2 • May 2024
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