A Simple Bias Reduction for Chatterjee’s Correlation

Chatterjee’s rank correlation coefficient ξ ₙ ξ n is an empirical index for detecting functional dependencies between two variables X and Y . It is an estimator for a theoretical quantity ξ ξ 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 ξ ₙ ξ 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 ξ ξ greater than about 0.4. Moreover, we observed that non-parametric confidence intervals for ξ ξ based on bootstrapping ξ ₙ ξ 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.