Title: Dependence and dependence structures: estimation and visualization using distance multivariance,
Abstract: 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'.