A Family of Chatterjee's Correlation Coefficients and Their Properties

Quantifying the strength of functional dependence between random scalars X and Y is a fundamental statistical problem.While many existing correlation coefficients effectively identify linear or monotone functional dependence, they often fail to capture more general, non-monotonic dependencies.In response, we propose a family of simple and computationally efficient correlation coefficients that encompasses a wide class of novel dependency measures, including the Chatterjee's correlation coefficient as a special case.We prove that each of our correlation coefficients converges to a population-level correlation measure as the sample size grows, and a central limit theorem is established under the assumption of independence between X and Y .Moreover, we provide the necessary and sufficient condition for the correlation measure to satisfy three desirable properties: (P1). it belongs to the range of [0, 1]; (P2). it equals 1 if and only if Y is a measurable function of X; and (P3). it equals 0 if and only if Y is independent of X.Based on both theoretical analysis and empirical evi-