Open Access
December 2023 S-estimation in linear models with structured covariance matrices
Hendrik Paul Lopuhaä, Valerie Gares, Anne Ruiz-Gazen
Author Affiliations +
Ann. Statist. 51(6): 2415-2439 (December 2023). DOI: 10.1214/23-AOS2334


We provide a unified approach to S-estimation in balanced linear models with structured covariance matrices. Of main interest are S-estimators for linear mixed effects models, but our approach also includes S-estimators in several other standard multivariate models, such as multiple regression, multivariate regression and multivariate location and scatter. We provide sufficient conditions for the existence of S-functionals and S-estimators, establish asymptotic properties such as consistency and asymptotic normality, and derive their robustness properties in terms of breakdown point and influence function. All the results are obtained for general identifiable covariance structures and are established under mild conditions on the distribution of the observations, which goes far beyond models with elliptically contoured densities. Some of our results are new and others are more general than existing ones in the literature. In this way, this manuscript completes and improves results on S-estimation in a wide variety of multivariate models. We illustrate our results by means of a simulation study and an application to data from a trial on the treatment of lead-exposed children.

Funding Statement

The third author was partially supported by by the French Agence Nationale de la Recherche through the Investments for the Future (Investissements d’Avenir) program, grant ANR-17-EURE-0010.


We thank both referees for pointing out relevant references and we thank the second referee whose comments lead to an improvement of Theorem 5.3.


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Hendrik Paul Lopuhaä. Valerie Gares. Anne Ruiz-Gazen. "S-estimation in linear models with structured covariance matrices." Ann. Statist. 51 (6) 2415 - 2439, December 2023.


Received: 1 June 2022; Revised: 1 October 2023; Published: December 2023
First available in Project Euclid: 20 December 2023

MathSciNet: MR4682703
zbMATH: 07783621
Digital Object Identifier: 10.1214/23-AOS2334

Primary: 62F12 , 62F35 , 62H12 , 62J05

Keywords: asymptotic distribution , Breakdown point , influence function , repeated measurements , S-estimators , S-functionals , structured covariance

Rights: This research was funded, in whole or in part, by [Agence National de la Recherche, ANR-17-EURE-0010]. A CC BY 4.0 license is applied to this article arising from this submission, in accordance with the grant's open access conditions

Vol.51 • No. 6 • December 2023
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