Asymptotics of estimators for structured covariance matrices

• Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands [h.p.lopuhaa@tudelft.nl]

Keywords: Linear model with structured covariances – S-estimators – Asymptotic variance – Influence function – MM-estimators

Abstract

We show that the limiting variance of a sequence of estimators for a structured covariance matrix has a general form, that for linear covariance structures appears as the variance of a scaled projection of a random matrix that is of radial type, and a similar result is obtained for the corresponding sequence of estimators for the vector of variance components. These results are illustrated by the limiting behavior of estimators for a differentiable covariance structure in a variety of multivariate statistical models. We also derive a characterization for the influence function of corresponding functionals.

Furthermore, we derive the limiting distribution and influence function of scale invariant mappings of such estimators and their corresponding functionals. As a consequence, the asymptotic relative efficiency of different estimators for the shape component of a structured covariance matrix can be compared by means of a single scalar and the gross error sensitivity of the corresponding influence functions can be compared by means of a single index. Similar results are obtained for estimators of the normalized vector of variance components.

Except that these results have a merit of their own, they also path the way for the construction of MM-estimators with auxiliary scale in linear mixed effects models and other linear models with structured covariances. These estimators inherit the robustness of S-estimators and improve both the efficiency of the estimator of the fixed effects as well as the efficiency of the estimator of the covariance shape component and of the direction of the vector of variance components. We discuss this version of MM-estimators, which will extend similar versions that are already available for unstructured covariances in the multivariate location-scale model, see Tatsuoka and Tyler [2000] or Salibián-Barrera et al. [2006], and in the multivariate regression model, see Kudraszow and Maronna [2011].