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

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.