Limit theorems of Azadkia-Chatterjee's conditional graph correlation

2026-06-13

AI Summary

This study addresses the absence of a statistical inference theory for the Azadkia–Chatterjee conditional dependence coefficient estimator \( T_n \) under general dependence structures. We establish its asymptotic normality and provide, for the first time, a central limit theorem valid for arbitrary dependence. By integrating rank-based and nearest-neighbor constructions, we derive a closed-form expression for the asymptotic variance and propose a consistent variance estimator computable in \( O(n \log n) \) time. Combined with existing bias-correction techniques, our results yield a complete inferential framework for this measure, substantially broadening its applicability in practical data analysis.

Abstract

Inferring the strength of conditional dependence and testing conditional independence are fundamental problems in statistics. A recent breakthrough by Azadkia and Chatterjee introduced, for the first time, a conditional dependence measure that equals $0$ if and only if the variables under study are conditionally independent, and equals $1$ if and only if they are conditionally perfectly dependent. They further proposed a computationally efficient and strongly consistent estimator, $T_n$, based on an ingenious use of ranks and nearest neighbors. Despite these attractive features, the asymptotic theory of $T_n$ has remained largely undeveloped. This paper closes that gap. We prove that, under general dependence, $T_n$ is asymptotically normal and its limiting variance admits a closed form. We also construct consistent variance estimators that are computationally efficient and implementable in $O(n\log n)$ time. Taken together with existing bias-correction methods, these results provide a complete inferential theory for $T_n$.

Problem

• conditional independence
• asymptotic normality
• variance estimation
• rank-based statistics
• nearest neighbors

Innovation

• conditional dependence
• asymptotic normality
• rank-based estimator
• nearest neighbors
• variance estimation