Nonparametric independence tests in metric spaces: What is known and what is not

TL;DR

This paper reviews the extension of distance correlation, a measure of independence, from Euclidean spaces to general metric spaces, summarizing current knowledge and identifying gaps for statisticians.

Contribution

It provides a comprehensive overview of the existing literature on nonparametric independence tests in metric spaces and offers original insights and proofs.

Findings

Distance correlation characterizes independence in Euclidean spaces.

Extensions to metric spaces are partially understood and under active development.

The review highlights key open problems and future research directions.

Abstract

Distance correlation is a recent extension of Pearson's correlation, that characterises general statistical independence between Euclidean-space-valued random variables, not only linear relations. This review delves into how and when distance correlation can be extended to metric spaces, combining the information that is available in the literature with some original remarks and proofs, in a way that is comprehensible for any mathematical statistician.