Let t(subl),...,t(sub K) be independent unbiased estimators of a parameter tau. Let t* = SUM alpha(sub i)t(sub i) have minimum variance among unbiased linear estimators of tau, and let t(hat) = SUM alpha(hat, sub i)t(sub i, SUM alpha(hat, sub i) = 1, be any linear combination with (alpha(hat,sub 1),..., alpha(hat, sub k)) independent of t(sub 1),...,t(sub k). We prove that t(hat) is unbiased for tau and that the ratio of the variance of t(hat) to the variance of t* is 1 + SUM MSE (alpha(hat, sub i))/alpha(sub i) where MSE (alpha(hat,sub i)) is the mean square error of alpha(hat, sub i) as an estimator of alpha(sub i). (7pp.)