Rearranged dependence measures

TL;DR

This paper introduces a novel method to transform existing dependence measures into ones that precisely characterize independence and functional dependence using monotone rearrangements.

Contribution

It proposes a general approach to modify dependence measures so they accurately reflect independence and functional relationships, applicable to measures like Spearman's and Kendall's.

Findings

Rearranged Spearman's ρ equals 0 only under independence.

Rearranged Kendall's τ equals 0 only under independence.

Proposed estimators are consistent and effective in finite samples.

Abstract

Most of the popular dependence measures for two random variables $X$ and $Y$ (such as Pearson's and Spearman's correlation, Kendall's $\tau$ and Gini's $\gamma$) vanish whenever $X$ and $Y$ are independent. However, neither does a vanishing dependence measure necessarily imply independence, nor does a measure equal to 1 imply that one variable is a measurable function of the other. Yet, both properties are natural properties for a convincing dependence measure. In this paper, we present a general approach to transforming a given dependence measure into a new one which exactly characterizes independence as well as functional dependence. Our approach uses the concept of monotone rearrangements as introduced by Hardy and Littlewood and is applicable to a broad class of measures. In particular, we are able to define a rearranged Spearman's $\rho$ and a rearranged Kendall's $\tau$ which do attain the value $0$ if and only if both variables are independent, and the value $1$ if and only if one variable is a measurable function of the other. We also present simple estimators for the rearranged dependence measures, prove their consistency and illustrate their finite sample properties by means of a simulation study and a data example.