A Simple Bias Reduction for Chatterjee's Correlation

Chatterjee's rank correlation coefficient xin 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 xin 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 xin 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.