May 2024 Rearranged dependence measures
Christopher Strothmann, Holger Dette, Karl Friedrich Siburg
Author Affiliations +
Bernoulli 30(2): 1055-1078 (May 2024). DOI: 10.3150/23-BEJ1624

Abstract

Most of the popular dependence measures for two random variables X and Y (such as Pearson’s and Spearman’s correlation, Kendall’s τ and Gini’s γ) 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 ρ and a rearranged Kendall’s τ 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.

Funding Statement

C. Strothmann gratefully acknowledges financial support from the German Academic Scholarship Foundation. The work of H. Dette was supported by the DFG Research Unit 5381 Mathematical Statistics in the Information Age project number 460867398.

Acknowledgements

The authors are grateful to two referees for their constructive comments on an earlier version of this paper.

Citation

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Christopher Strothmann. Holger Dette. Karl Friedrich Siburg. "Rearranged dependence measures." Bernoulli 30 (2) 1055 - 1078, May 2024. https://doi.org/10.3150/23-BEJ1624

Information

Received: 1 August 2022; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699545
Digital Object Identifier: 10.3150/23-BEJ1624

Keywords: coefficient of correlation , copula , decreasing rearrangement , measure of dependence

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Vol.30 • No. 2 • May 2024
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