Bias correction for Chatterjee’s graph-based correlation coefficient

Abstract

Azadkia & Chatterjee (2021) recently introduced a simple nearest-neighbour graph-based correlation coefficient that consistently detects both independence and functional dependence, in both unconditional and conditional settings. Specifically, it approximates a measure of dependence that equals 0 if and only if the variables are (conditionally) independent, and 1 if and only if they are (conditionally) functionally dependent. However, this nearest-neighbour estimator includes a bias term that may vanish at a rate slower thann1/2, preventing consistency at raten1/2in general. In this article, focusing on the unconditional version, we (i) analyse this bias term closely and show that it could become asymptotically negligible when the dimension is at most three and (ii) propose a bias-correction procedure for more general settings. In both regimes, we obtain estimators (either the original or the bias-corrected version) that are consistent at the parametric raten1/2and asymptotically normal.