A direct extension of Azadkia & Chatterjee's rank correlation to multi-response vectors

Recently, Chatterjee (2023) recognized the lack of a direct generalization of his rank correlation xi in Azadkia and Chatterjee (2021) to a multi-dimensional response vector. As a natural solution to this problem, we here propose an extension of xi that is applicable to a set of q ≥ 1 response variables, where our approach builds upon converting the original vector-valued problem into a univariate problem and then applying the rank correlation xi to it. Our novel measure T quantifies the scale-invariant extent of functional dependence of a response vector Y = (Y1,dots,Yq) on predictor variables X = (X1, dots,Xp), characterizes independence of X and Y as well as perfect dependence of Y on X and hence fulfills all the characteristics of a measure of predictability. Aiming at maximum interpretability, we provide various invariance results for T as well as a closed-form expression in multivariate normal models. Building upon the graph-based estimator for xi in Azadkia and Chatterjee (2021), we obtain a non-parametric, strongly consistent estimator for T and show -- as a main contribution -- its asymptotic normality. Based on this estimator, we develop a model-free and rank-based feature ranking and forward feature selection for multiple-outcome data that works without any tuning parameters. Simulation results and real case studies illustrate T's broad applicability.