Bias correction for Chatterjee’s graph-based correlation coefficient

Summary 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 than n1/2, preventing consistency at rate n1/2 in 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 rate n1/2 and asymptotically normal.