A bivariate copula capturing the dependence of a random variable and a random vector, its estimation and applications
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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, …, X_d. The copula itself contains the information whether Y is completely dependent on X_1, …, X_d, and whether Y and X_1, …, 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.
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