October 2022 Affine-equivariant inference for multivariate location under Lp loss functions
Alexander Dürre, Davy Paindaveine
Author Affiliations +
Ann. Statist. 50(5): 2616-2640 (October 2022). DOI: 10.1214/22-AOS2199


We consider the fundamental problem of estimating the location of a d-variate probability measure under an Lp loss function. The naive estimator, that minimizes the usual empirical Lp risk, has a known asymptotic behavior but suffers from several deficiencies for p2, the most important one being the lack of equivariance under general affine transformations. In this work, we introduce a collection of Lp location estimators μˆnp, that minimize the size of suitable -dimensional data-based simplices. For =1, these estimators reduce to the naive ones, whereas, for =d, they are equivariant under affine transformations. Irrespective of , these estimators reduce to the sample mean for p=2, whereas for p=1, the estimators provide the well-known spatial median and Oja median for =1 and =d, respectively. Under very mild assumptions, we derive an explicit Bahadur representation result for μˆnp, and establish asymptotic normality. We prove that, quite remarkably, the asymptotic behavior of the estimators does not depend on under spherical symmetry, so that the affine equivariance for =d is achieved at no cost in terms of efficiency. To allow for large sample size n and/or large dimension d, we introduce a version of our estimators relying on incomplete U-statistics. Under a centro-symmetry assumption, we also define companion tests ϕnp, for the problem of testing the null hypothesis that the location μ of the underlying probability measure coincides with a given location μ0. For any p, affine invariance is achieved for =d. For any and p, we derive explicit expressions for the asymptotic power of these tests under contiguous local alternatives, which reveals that asymptotic relative efficiencies with respect to traditional parametric Gaussian procedures for hypothesis testing coincide with those obtained for point estimation. We illustrate finite-sample relevance of our asymptotic results through Monte Carlo exercises and also treat a real data example.

Funding Statement

This research was supported the Program of Concerted Research Actions (ARC) of the Université libre de Bruxelles.


We would like to thank the Editor, the Associate Editor and two anonymous referees for their insightful comments and suggestions that led to a substantial improvement of a previous version of this work.


Download Citation

Alexander Dürre. Davy Paindaveine. "Affine-equivariant inference for multivariate location under Lp loss functions." Ann. Statist. 50 (5) 2616 - 2640, October 2022. https://doi.org/10.1214/22-AOS2199


Received: 1 September 2021; Revised: 1 February 2022; Published: October 2022
First available in Project Euclid: 27 October 2022

MathSciNet: MR4500620
zbMATH: 07628834
Digital Object Identifier: 10.1214/22-AOS2199

Primary: 62G05 , 62G20
Secondary: 60D05 , 62H15

Keywords: Affine equivariance/affine invariance , contiguous alternatives , Lp loss functions , multivariate location functionals , Oja median , Random simplices , spatial median

Rights: Copyright © 2022 Institute of Mathematical Statistics


This article is only available to subscribers.
It is not available for individual sale.

Vol.50 • No. 5 • October 2022
Back to Top