What question did this study set out to answer?

This research focuses on developing generalized least squares estimators for analyzing covariance structures and assessing their asymptotic properties.

June 1, 1973Open Access

Generalized Least Squares Estimators in the Analysis of Covariance Structures

Puntos clave

This research focuses on developing generalized least squares estimators for analyzing covariance structures and assessing their asymptotic properties.
Minimization of a positive definite matrix to obtain G.L.S. estimates.
Investigation of asymptotic properties under regularity conditions and multivariate normal distributions.
Calculation of estimates and test statistics for various linear models.
The M.W.L. estimator is identified as a G.L.S. estimator with minimum asymptotic variance.
G.L.S. estimators exhibit less computational complexity compared to traditional methods.
Provided methods enhance the estimation of dispersion matrices within linear models.

Resumen

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.

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