Studying the multivariate extension of the copula correlation yields the principle of dimensionality reduction. This was found to be strongly related to the “simple measure of conditional dependence” $T$ recently introduced by Azadkia & Chatterjee (2021). In this paper, we identify and investigate the dependence structure underlying this dimensionality reduction principle, provide it with a strong and consistent estimator, and demonstrate its broad applicability. For that purpose, $d \geq 1$ exogenous random variable ${\bf X} = (X_1, \dots , X_d)$ of endogenous random variable $Y$, and $Y$ is ${\bf Contains information on whether X}$ is fully dependent and whether $Y$ and ${\bf X}$ are independent. The bivariate copula introduced is two random variables $Y$ and $Y^\prime$ that share the same conditional distribution and are conditionally independent, given ${\bf X}$ As long as we can consider the distribution function of , the principle of dimensionality reduction becomes clear. Evaluating this copula uniformly along the diagonal, i.e. computing Spearman’s foot rule, yields Azadkia and Chatterjee’s “conditional simple measure” $T$. On the other hand, evaluating this copula uniformly over the unit square, or computing Spearman’s rho, yields the undistributed coefficient of determination (a.k.a. copula correlation). Several real data examples demonstrate the importance of the introduced methodology.
