Projection estimators for unknown covariance matrices in linear models

Abstract

The estimation or approximation of the unknown covariance matrix with linear structure in a linear model may be performed by estimation of the linear covariance parameter with any accessible estimator like MLE, MINQUE, Bique or Baique. The paper presents another approach, which consists in the approximation by a matrix with very simple linear structure, e.g. by σ² I, and by a convenient choice of the linear parameter essentially by projection of the estimated covariance matrix into the linear space given by the simple structure. The risk function for such a procedure is investigated and compaired with the risk for the other approximations. Special cases of heteroscedastic variances and of equicorrelated observations are used as examples, where usual estimators are improved uniformly by projection estimators.