S-estimation in Linear Models with Structured Covariance Matrices

TL;DR

This paper develops a comprehensive framework for S-estimation in balanced linear models with structured covariance matrices, covering various multivariate models and establishing their theoretical properties and robustness.

Contribution

It introduces new conditions and generalizes existing results for S-estimators across multiple multivariate models with structured covariance matrices.

Findings

Established existence conditions for S-estimators.

Proved asymptotic consistency and normality.

Demonstrated robustness through simulation and real data application.

Abstract

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.