Symmetric Rank Covariances: a Generalised Framework for Nonparametric Measures of Dependence

TL;DR

This paper introduces Symmetric Rank Covariances, a unified framework for nonparametric dependence measures that are invariant under transformations, generalizing existing measures and providing efficient algorithms for their computation.

Contribution

It defines a new class of dependence measures called Symmetric Rank Covariances, unifying and extending existing measures like Kendall's tau and Hoeffding's D, with efficient computation methods.

Findings

Unified framework for dependence measures

Generalization of multivariate sign covariance

Efficient algorithms for Hoeffding's D in multivariate setting

Abstract

The need to test whether two random vectors are independent has spawned a large number of competing measures of dependence. We are interested in nonparametric measures that are invariant under strictly increasing transformations, such as Kendall's tau, Hoeffding's D, and the more recently discovered Bergsma--Dassios sign covariance. Each of these measures exhibits symmetries that are not readily apparent from their definitions. Making these symmetries explicit, we define a new class of multivariate nonparametric measures of dependence that we refer to as Symmetric Rank Covariances. This new class generalises all of the above measures and leads naturally to multivariate extensions of the Bergsma--Dassios sign covariance. Symmetric Rank Covariances may be estimated unbiasedly using U-statistics for which we prove results on computational efficiency and large-sample behavior. The algorithms we develop for their computation include, to the best of our knowledge, the first efficient algorithms for the well-known Hoeffding's D statistic in the multivariate setting.