THE BEST QUADRATIC MINIMUM BIAST NONNEGATIVE DEFINITE ESTIMATOR FOR AN ADDITIVE 2 VARIANCE COMPONENT MODEL

We derive the Best Quadratic Minimum Bias Non-negative Definite Estima tors of the variance components sigma(1)(2) and sigma(2)(2) of the var iance-covariance matrix model Q = sigma(1)(2)I + sigma(2)(2)F for the observation vector L in the linear model AX = E{L}. I is the unit matr ix and F is a positive diagonal matrix. The result is [GRAPHICS] and [ GRAPHICS] Here f(min) and f(max) are the minimum and maximum elements of F, and <(epsilon)over cap>(i) and <(epsilon)over cap>(j) the corres ponding residuals obtained in the least squares adjustment. In additio n to the design matrix A we introduce nu = sigma(2)/sigma(1) and A(O) = A(A(T)Q(O)(-1)A)(-1) A(T) Q(O)(-1), where Q(O) is the estimate for Q given by a priori presented components sigma(1) and sigma(2). As each estimator is determined from merely one residual, it turns out that t heir variances are poor.