A framework for measuring dependence between random vectors

TL;DR

This paper introduces a flexible framework for measuring dependence between random vectors using collapsing functions, enabling graphical assessments, new dependence measures, and applications in bioinformatics and finance.

Contribution

It proposes a novel collapsing function framework, extends Kendall distribution and copula to multivariate cases, and develops non-parametric estimators with asymptotic properties.

Findings

Graphical independence assessment for random vectors.

Extension of Kendall distribution and copula to multivariate dependence.

Non-parametric estimators with proven asymptotic properties.

Abstract

A framework for quantifying dependence between random vectors is introduced. With the notion of a collapsing function, random vectors are summarized by single random variables, called collapsed random variables in the framework. Using this framework, a general graphical assessment of independence between groups of random variables for arbitrary collapsing functions is provided. Measures of association computed from the collapsed random variables are then used to measure the dependence between random vectors. To this end, suitable collapsing functions are presented. Furthermore, the notion of a collapsed distribution function and collapsed copula are introduced and investigated for certain collapsing functions. This investigation yields a multivariate extension of the Kendall distribution and its corresponding Kendall copula for which some properties and examples are provided. In addition, non-parametric estimators for the collapsed measures of dependence are provided along with their corresponding asymptotic properties. Finally, data applications to bioinformatics and finance are presented.