The Distance Standard Deviation

Dominic Edelmann, Donald Richards, Daniel Vogel

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)3395-3416
Number of pages24
JournalAnnals of Statistics
Issue number6
Early online date11 Dec 2020
Publication statusPublished - 11 Dec 2020

Bibliographical note

Supplement to “The distance standard deviation”. The supplementary material consists of six appendices. Appendix A contains various proofs to results in the main paper. Appendix B contains details on the derivation of Table 1 in the main document. In Appendix C, a limit theorem for the squared distance covariance is stated, under weaker assumptions than known previously. Simulation results for permutation-based two-sample scale tests are provided in Appendix D. Appendix E gives additional theoretical results for the distance standard deviation in one dimension. Appendix F tabulates the distance variances for a collection of distributions.


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