LEAST-SQUARES LEHMANN-SCHEFFE ESTIMATION OF VARIANCES AND COVARIANCESWITH MIXED LINEAR-MODELS

Variances of quadratic estimators of (co)variances are functions of th e numeric values of the (co)variance parameters being estimated. This situation makes estimation of (co)variances problematical. Uniformly b est quadratic, unbiased estimators exist for balanced designs but not for unbalanced designs. This article tackles the problem by providing explicit quadratic estimators of(co)variances that are uniformly best in the sense that they are uniformly minimum variance, unbiased to the maximum extent possible over the entire range of possible parameter v alues of the (co)variances being estimated. This was accomplished by d etermining the restrictions on the elements of the matrix of the quadr atic-form matrix necessary to satisfy the Lehmann-Scheffe criterion fo r uniformly minimum variance, unbiased estimation and then solving the resulting linear equations via the principle of least squares. The co ntext is any mixed linear model, and the approach does not require tha t there be equal numbers of observations in the case of multivariate d ata. A detailed development of the method is given. That the procedure is completely general is discussed. A modification that forces unbias edness is presented. An example with a three-variance-component model is provided and results discussed. A miscellaneous section discusses, among other topics, how this method can be used to compare other (co)v ariance component estimation procedures. The final section illustrates how the method handles multivariate situations (i.e., models with bot h variances and covariances) by detailing the expressions involved wit h the bivariate model.