Printed in Great Britain Optimality of Royall's predictor under a Gaussian superpopulation model

This note shows that under the assumption of a Gaussian superpopulation model with a general symmetric and positive-definite covariance matrix, the model-based predictor of Royall (1976) is the unbiased minimum mean squared error predictor of the total of a given finite population. This somewhat anticipated result is a generalization of recent published work in which it was derived under more restrictive assumptions. Some key words: Minimum mean squared error predictor; Model-based approach; Survey sampling. In this note, we assume that the finite population of interest y ' = (yt,..., yN) is a realization of a random vector Y ' which is related to the matrix X = (x, ,..., xN) ' by Y = Xfi + e, where X is assumed of full rank, e is Gaussian with a null mean vector and covariance matrix Va2, V is a known symmetric positive-definite matrix, and a and the p x 1 vector)9 are unknown. We are interested in inferring the population total, VY, based on a sample of n population units, where 1 denotes an N x 1 vector of ones. The following uses the same notation as Tarn (1986). Let /(Y,) be a predictor of 1 ' Y. It is true that-f(Y,)}2 = E{l'Y-E(l'Y\Y,)}2+E{f(Y,)-E(l'Y\Y,)}2, (1) from which it follows that the minimum mean squared error predictor of \'Y is the minimum mean squared error predictor of £ ( l ' y | Y,), and vice versa