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December 2020 The distance standard deviation
Dominic Edelmann, Donald Richards, Daniel Vogel
Ann. Statist. 48(6): 3395-3416 (December 2020). DOI: 10.1214/19-AOS1935


The distance standard deviation, which arises in distance correlation analysis of multivariate data, is studied as a measure of spread. The asymptotic distribution of the empirical distance standard deviation is derived under the assumption of finite second moments. Applications are provided to hypothesis testing on a data set from materials science and to multivariate statistical quality control. The distance standard deviation is compared to classical scale measures for inference on the spread of heavy-tailed distributions. Inequalities for the distance variance are derived, proving that the distance standard deviation is bounded above by the classical standard deviation and by Gini’s mean difference. New expressions for the distance standard deviation are obtained in terms of Gini’s mean difference and the moments of spacings of order statistics. It is also shown that the distance standard deviation satisfies the axiomatic properties of a measure of spread.


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Dominic Edelmann. Donald Richards. Daniel Vogel. "The distance standard deviation." Ann. Statist. 48 (6) 3395 - 3416, December 2020.


Received: 1 November 2018; Revised: 1 November 2019; Published: December 2020
First available in Project Euclid: 11 December 2020

MathSciNet: MR4185813
Digital Object Identifier: 10.1214/19-AOS1935

Primary: 60E15 , 62H20
Secondary: 60E05 , 60E10

Keywords: Asymptotic efficiency , Characteristic function , dispersive ordering , distance correlation coefficient , distance variance , Gini’s mean difference , measure of spread , order statistic , sample spacing , stochastic ordering , U-statistic

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 6 • December 2020
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