A bivariate copula capturing the dependence of a random variable and a random vector, its estimation and applications

We define a bivariate copula that captures the scale-invariant extent of dependence of a single random variable $Y$ on a set of potential explanatory random variables $X_1, \dots, X_d$. The copula itself contains the information whether $Y$ is completely dependent on $X_1, \dots, X_d$, and whether $Y$ and $X_1, \dots, X_d$ are independent. Evaluating this copula uniformly along the diagonal, i.e. calculating Spearman's footrule, leads to the so-called 'simple measure of conditional dependence' recently introduced by Azadkia and Chatterjee [1]. On the other hand, evaluating this copula uniformly over the unit square, i.e. calculating Spearman's rho, leads to a distribution-free coefficient of determination. Applying the techniques introduced in [1], we construct an estimate for this copula and show that this copula estimator is strongly consistent. Since, for $d=1$, the copula under consideration coincides with the well-known Markov product of copulas, as by-product, we also obtain a strongly consistent copula estimator for the Markov product. A simulation study illustrates the small sample performance of the proposed estimator.