A family of Chatterjee's correlation coefficients and their properties

Quantifying the strength of functional dependence between random scalars X and Y is an important statistical problem. While many existing correlation coefficients excel in identifying linear or monotone functional dependence, they fall short in capturing general non-monotone functional relationships. In response, we propose a family of correlation coefficients ξ(h,F)ₙ, characterized by a continuous bivariate function h and a cdf function F. By offering a range of selections for h and F, ξ(h,F)ₙ encompasses a diverse class of novel correlation coefficients, while also incorporates the Chatterjee's correlation coefficient (Chatterjee, 2021) as a special case. We prove that ξ(h,F)ₙ converges almost surely to a deterministic limit ξ(h,F) as sample size n approaches infinity. In addition, under appropriate conditions imposed on h and F, the limit ξ(h,F) satisfies the three appealing 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. As amplified by our numerical experiments, our proposals provide practitioners with a variety of options to choose the most suitable correlation coefficient tailored to their specific practical needs.