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 AV_i = p_i, i = 1,.., k, or U₁'AU₁ + ... + U_k'AU_k = 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.