Multivariate MM-estimators with auxiliary Scale for Linear Models with Structured Covariance Matrices

Abstract: We provide a unified approach to MM-estimation with auxiliary scale for balanced linear models with structured covariance matrices. This approach leads to estimators that are highly robust against outliers and highly efficient for normal data. These properties not only hold for estimators of the regression parameter, but also for estimators of scale invariant transformations of the variance parameters. Of main interest are MM-estimators for linear mixed effects models, but our approach also includes MM-estimators in several other standard multivariate models. We provide sufficient conditions for the existence of MM-functionals and MM-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.