Dependence and dependence structures: estimation and visualization using distance multivariance

Distance multivariance is a multivariate dependence measure, which can detect dependencies between an arbitrary number of random vectors each of which can have a distinct dimension. Here we discuss several new aspects and present a concise overview. We relax the required moment conditions considerably and show that distance multivariance unifies (and extends) distance covariance and the Hilbert-Schmidt independence criterion HSIC, moreover also the classical linear dependence measures: covariance, Pearson's correlation and the RV coefficient appear as limiting cases. For measures based on distance multivariance the corresponding resampling tests are introduced, and several related measures are defined: a new multicorrelation which satisfies a natural set of multivariate dependence measure axioms and $m$-multivariance which is a new dependence measure yielding tests for pairwise independence and independence of higher order. These tests are computationally feasible and under very mild moment conditions they are consistent against all alternatives. Moreover, a general visualization scheme for higher order dependencies is proposed. Many illustrative examples are included. All functions for the use of distance multivariance in applications are published in the R-package 'multivariance'.