A Simple Bias Reduction for Chatterjee's Correlation

TL;DR

This paper introduces a simple normalization technique for Chatterjee's correlation coefficient to reduce bias in small samples, improving accuracy and confidence interval coverage.

Contribution

The authors propose a new normalization method for Chatterjee's correlation that reduces bias and mean squared error in small samples, and improve bootstrap confidence interval coverage.

Findings

Bias reduction in small samples across various models

Decreased mean squared error for higher dependency values

Improved bootstrap confidence interval coverage

Abstract

Chatterjee's rank correlation coefficient $\xi_n$ is an empirical index for detecting functional dependencies between two variables $X$ and $Y$. It is an estimator for a theoretical quantity $\xi$ that is zero for independence and one if $Y$ is a measurable function of $X$. Based on an equivalent characterization of sorted numbers, we derive an upper bound for $\xi_n$ and suggest a simple normalization aimed at reducing its bias for small sample size $n$. In Monte Carlo simulations of various models, the normalization reduced the bias in all cases. The mean squared error was reduced, too, for values of $\xi$ greater than about 0.4. Moreover, we observed that non-parametric confidence intervals for $\xi$ based on bootstrapping $\xi_n$ in the usual n-out-of-n way have a coverage probability close to zero. This is remedied by an m-out-of-n bootstrap without replacement in combination with our normalization method.