Highly Efficient Estimators with High Breakdown Point for Linear Models with Structured Covariance Matrices

TL;DR

This paper introduces a unified estimation approach for linear models with structured covariance matrices that achieves high robustness and efficiency, applicable to various multivariate models including linear mixed effects.

Contribution

It provides new and generalized estimators with high breakdown points and efficiency, along with theoretical guarantees for their robustness and asymptotic properties.

Findings

Establishes conditions for estimator existence and properties.

Demonstrates robustness through breakdown point and influence function analysis.

Extends results to general covariance structures beyond elliptically contoured models.

Abstract

We provide a unified approach to a method of estimation of the regression parameter in balanced linear models with a structured covariance matrix that combines a high breakdown point and bounded influence with high asymptotic efficiency at models with multivariate normal errors. Of main interest are linear mixed effects models, but our approach also includes several other standard multivariate models, such as multiple regression, multivariate regression, and multivariate location and scatter. We provide sufficient conditions for the existence of the estimators and corresponding functionals, 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 high breakdown estimation with high efficiency in a wide variety of multivariate models.