Generalized Least Squares Estimators in the Analysis of Covariance Structures

SUMMARY Let S represent the usual unbiased estimator of a covariance matrix, Σ 0 , whose elements are functions of a parameter vector . A generalized least squares (G.L.S) estimate, of may be obtained by minimizing where V is some positive definite matrix. Asymptotic properties of the G.L.S. estimators are investigated assuming only that satisfies certain regularity conditions and that the limiting distribution of S is multivariate normal with specified parameters. The estimator of which is obtained by maximizing the Wishart likelihood function (M.W.L. estimator) is shown to be a member of the class of G.L.S. estimators with minimum asymptotic variances. When is linear in a G.L.S. estimator which converges stochastically to the M.W.L. estimator involves far less computation. Methods for calculating estimates of , estimates of the dispersion matrix of , and test statistics, are given for certain linear models.