Least squares Lehmann-Scheffé estimation of variances and covariances with mixed linear models.

Variances of quadratic estimators of (co)variances are functions of the numeric values of the (co)variance parameters being estimated. This situation makes estimation of (co)variances problematical. Uniformly best 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 values of the (co)variances being estimated. This was accomplished by determining the restrictions on the elements of the matrix of the quadratic-form matrix necessary to satisfy the Lehmann-Scheffé criterion for uniformly minimum variance, unbiased estimation and then solving the resulting linear equations via the principle of least squares. The context is any mixed linear model, and the approach does not require that there be equal numbers of observations in the case of multivariate data. A detailed development of the method is given. That the procedure is completely general is discussed. A modification that forces unbiasedness 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)variance component estimation procedures. The final section illustrates how the method handles multivariate situations (i.e., models with both variances and covariances) by detailing the expressions involved with the bivariate model.