Residual Optimality: Ordinary Vs. Weighted Vs. Biased Least Squares

In the general linear model with observations not necessarily uncorrelated or homoscedastic, Gauss-Markov regression coefficients are superior to ordinary unweighted least squares in the well known BLU sense if the model is correct. However, it is shown that there is a weaker, but always applicable,...

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Bibliographic Details
Published in:Journal of the American Statistical Association 1975-06, Vol.70 (350), p.375-379
Main Author: Obenchain, R. L.
Format: Article
Language:English
Subjects:
Covariance matrices
Eigenvalues
Estimation bias
Estimators
General linear model
Least squares
Mathematical vectors
Remainders
Statistical variance
Theory and Methods
Unbiased estimators
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Summary:In the general linear model with observations not necessarily uncorrelated or homoscedastic, Gauss-Markov regression coefficients are superior to ordinary unweighted least squares in the well known BLU sense if the model is correct. However, it is shown that there is a weaker, but always applicable, minimum overall mean squared error sense in which Gauss-Markov residuals and biased residuals are inferior to ordinary least squares residuals as estimators of possible lack-of-fit in the model. This optimality of ordinary least squares is further illustrated by three other types of results about residuals.
ISSN:0162-1459
1537-274X
DOI:10.1080/01621459.1975.10479876