A New Class of Nonsymmetric Multivariate Dependence Measures

TL;DR

This paper introduces a new class of copula-based, nonsymmetric multivariate dependence measures that characterize various dependence structures, extend existing mathematical frameworks, and are invariant under transformations.

Contribution

It proposes a novel set of conditions for multivariate dependence measures, constructs explicit measures satisfying these, and extends the star product to multivariate copulas.

Findings

Measures satisfy DPI and self-equitability.

Extension of dependence measures to groups of variables.

Invariance under continuous bijective transformations.

Abstract

Following our previous work on copula-based nonsymmetric bivariate dependence measures, we propose a new set of conditions on nonsymmetric multivariate dependence measures which characterize both independence and complete dependence of one random variable on a group of random variables. The measures are nonparametric in that they are copula-based and are invariant under continuous bijective transformations on the group of random variables. We also construct explicitly new measures that satisfy the conditions. Besides, we extend the star product on bivariate copulas to multivariate copulas and prove the DPI condition and self-equitability for the new measures. A further extension to measures of dependence of one group of random variables on another group of random variables is also discussed.