On a multivariate copula-based dependence measure and its estimation

Abstract

Working with so-called linkages allows to define a copula-based, [0, 1]-valued multivariate dependence measure ζ1 (X, Y) quantifying the scale-invariant extent of dependence of a random variable Y on a d-dimen-sional random vector X = (X1, …, Xd) which exhibits various good and natural properties. In particular, ζ1 (X, Y) = 0 if and only if X and Y are independent, ζ1 (X, Y) is maximal exclusively if Y is a function of X, and ignoring one or several coordinates of X can not increase the resulting dependence value. After introducing and analyzing the metric D1 underlying the construction of the dependence measure and deriving examples showing how much information can be lost by only considering all pairwise dependence values ζ1 (X1, Y), …, ζ1 (Xd, Y) we derive a so-called checker-board estimator for ζ1 (X, Y) and show that it is strongly consistent in full generality, i.e., without any smoothness restrictions on the underlying copula. Some simulations illustrating the small sample performance of the estimator complement the established theoretical results.