On Nonsymmetric Nonparametric Measures of Dependence

TL;DR

This paper introduces a new class of nonsymmetric dependence measures based on copulas, which better capture relationships between variables and satisfy desirable properties like the Data Processing Inequality.

Contribution

It proposes a new set of axioms for nonsymmetric dependence measures, extending Renyi's axioms, and demonstrates their advantages over symmetric measures.

Findings

Nonsymmetric measures better characterize variable relationships.

New measures satisfy Data Processing Inequality.

Invariance under bijective transformations achieved.

Abstract

Based on recent progress in research on copula based dependence measures, we review the original Renyi's axioms on symmetric measures and propose a new set of axioms that applies to nonsymmetric measures. We show that nonsymmetric measures can actually better characterize the relationship between a pair of random variables including both independence and complete dependence. The new measures also satisfy the Data Processing Inequality (DPI) on the * product on copulas, which leads to nice features including the invariance of dependence measure under bijective transformation on one of the random variables. The issues with symmetric measures are also clarified.