Exact Detection Thresholds for Chatterjee's Correlation

Recently, Chatterjee (2021) introduced a new rank-based correlation coefficient which can be used to test for independence between two random variables. His test has already attracted much attention as it is distribution-free, consistent against all fixed alternatives, asymptotically normal under the null hypothesis of independence and computable in (near) linear time; thereby making it appropriate for large-scale applications. However, not much is known about the power properties of this test beyond consistency against fixed alternatives. In this paper, we bridge this gap by obtaining the asymptotic distribution of Chatterjee's correlation under any changing sequence of alternatives "converging" to the null hypothesis (of independence). We further obtain a general result that gives exact detection thresholds and limiting power for Chatterjee's test of independence under natural nonparametric alternatives "converging" to the null. As applications of this general result, we prove a non-standard $n^{-1/4}$ detection boundary for this test and compute explicitly the limiting local power on the detection boundary, for popularly studied alternatives in literature such as mixture models, rotation models and noisy nonparametric regression. Moreover our convergence results provide explicit finite sample bounds that depend on the "distance" between the null and the alternative. Our proof techniques rely on second order Poincar\'{e} type inequalities and a non-asymptotic projection theorem.