Asymptotics of estimators for structured covariance matrices

TL;DR

This paper derives the asymptotic behavior, influence functions, and efficiency comparisons of estimators for structured covariance matrices, providing a unified framework for analyzing their statistical properties in multivariate models.

Contribution

It introduces a general form for the limiting variance of structured covariance estimators and characterizes their influence functions, enabling comprehensive efficiency and robustness analysis.

Findings

Limiting variance expressed as a scaled projection of a radial-type random matrix

Derived influence functions for covariance estimators and their functionals

Compared efficiency and robustness of estimators using scalar indices

Abstract

We show that the limiting variance of a sequence of estimators for a structured covariance matrix has a general form that 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 linear 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. We apply our results to investigate how the efficiency, gross error sensitivity, and breakdown point of S-estimators for the normalized variance components are affected simultaneously by varying their cutoff value.