Open Access
Translator Disclaimer
May 2022 On the measure of anchored Gaussian simplices, with applications to multivariate medians
Davy Paindaveine
Author Affiliations +
Bernoulli 28(2): 965-996 (May 2022). DOI: 10.3150/21-BEJ1373

Abstract

We consider anchored Gaussian -simplices in the d-dimensional Euclidean space, that is, simplices with one fixed vertex yRd and the remaining vertices X1,,X randomly sampled from the d-variate standard normal distribution. We determine the distribution of the measure of such simplices for any d, any , and any anchor point y, which is of interest, e.g., when studying the asymptotic behaviour of U-statistics based on such simplex measures. We provide two proofs of the results. The first one is short but is not self-contained as it crucially relies on a technical result for non-central Wishart distributions. The second one is a simple and self-contained proof, that also provides some geometric insight on the results. Quite nicely, variations on this second argument reveal intriguing distributional identities on products of central and non-central chi-square distributions with Beta-distributed non-centrality parameters. We independently establish these distributional identities by making use of Mellin transforms. Beyond the aforementioned use to study the asymptotic behaviour of some U-statistics, our results do find natural applications in the context of robust location estimation, as we illustrate by considering a class of simplex-based multivariate medians that contains the celebrated spatial median and Oja median as special cases. Throughout, our results are confirmed by numerical experiments.

Funding Statement

This research work was supported by a research fellowship from the Francqui Foundation and by the Program of Concerted Research Actions (ARC) of the Université libre de Bruxelles.

Acknowledgements

The author would like to thank the Editor-In-Chief, Mark Podolskij, the Associate Editor, and three anonymous referees for their insightful comments and suggestions.

Citation

Download Citation

Davy Paindaveine. "On the measure of anchored Gaussian simplices, with applications to multivariate medians." Bernoulli 28 (2) 965 - 996, May 2022. https://doi.org/10.3150/21-BEJ1373

Information

Received: 1 January 2021; Revised: 1 May 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

Digital Object Identifier: 10.3150/21-BEJ1373

Keywords: distributional identities , Mellin transforms , Multivariate medians , Oja median , Random simplices , spatial median , Stochastic geometry

Rights: Copyright © 2022 ISI/BS

JOURNAL ARTICLE
32 PAGES


SHARE
Vol.28 • No. 2 • May 2022
Back to Top