Screening and Monotonic Dependence Functions in the Multivariate Case

Abstract

An approach to simultaneous treatment of dependence and screening problems is presented. New characterizations of dependence of a random variable $X$ on a random vector $Y$ are obtained by functions $\nu_{X, Y}: (0, 1)\rightarrow \lbrack 0, 1\rbrack$ and $\mu_{X, Y} : (0, 1) \rightarrow \lbrack -1, 1\rbrack$ called respectively screening and monotonic dependence functions. These functions are shown to be appropriate measures of the intensity of connection and concordance of $X$ on $Y$, respectively. The interrelations of $\nu$ and $\mu$ and their relations to the multiple correlation ratio and the multiple correlation coefficient are demonstrated and illustrated by several examples.