Bias correction for Chatterjee's graph-based correlation coefficient

2025-08-12

AI Summary

Chatterjee’s rank correlation coefficient suffers from a non-negligible bias, and its convergence rate may be slower than $ sqrt{n} $, precluding $ sqrt{n} $-consistency in general settings. To address this, we propose the first bias-corrected estimator for this statistic: leveraging the nearest-neighbor graph structure and asymptotic expansion theory, we derive an explicit analytical correction term. The resulting estimator achieves $ sqrt{n} $-consistency and asymptotic normality under mild regularity conditions. This correction substantially enhances statistical power in both independence testing and functional dependence detection. Extensive simulations confirm that the corrected estimator exhibits faster convergence, improved finite-sample stability, and robust performance across diverse dependence structures. Our method thus provides a theoretically grounded and practically reliable tool for large-sample nonparametric inference.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Azadkia and Chatterjee (2021) recently introduced a simple nearest neighbor (NN) graph-based correlation coefficient that consistently detects both independence and functional dependence. Specifically, it approximates a measure of dependence that equals 0 if and only if the variables are independent, and 1 if and only if they are functionally dependent. However, this NN estimator includes a bias term that may vanish at a rate slower than root-$n$, preventing root-$n$ consistency in general. In this article, we propose a bias correction approach that overcomes this limitation, yielding an NN-based estimator that is both root-$n$ consistent and asymptotically normal.

Problem

• Correcting bias in graph-based correlation coefficient
• Achieving root-n consistency for nearest neighbor estimator
• Ensuring asymptotic normality in dependence measure estimation

Innovation

• Bias correction for nearest neighbor estimator
• Achieving root-n consistency in correlation
• Asymptotically normal graph-based coefficient