Estimation of variance and covariance components—MINQUE theory

Abstract

The paper consists of two parts. The first part deals with solutions to some optimization problems. The general problem is one of minimssing Tr AVA′U, where V and U are positive definite matrices when the elements of the matrix are subject to linear restrictions of the type AX = O or X′AX = O and trace AVi = pi, i = 1,…, k, or U1′AU1 + … + Uk′AUk = M. These results are used in determining Minimum Norm Quadratic Unbiased Estimators (MINQUE) of variance and covariance components in linear models. The present paper is a generalization of an earlier attempt by the author to obtain estimators of heteroscedastic variances in a regression model. The method is quite general, applicable to all experimental situations, and the computations are simple.