Uncorrelated residuals from linear models |
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Authors: | Warren T Dent George PH Styan |
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Institution: | University of Iowa, Iowa City, IA 52242, USA;McGill University, Montreal, Quebec, Canada |
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Abstract: | In the usual linear model , the error vector u is not observable and the vector r of least squares residuals has a singular covariance matrix that depends on the design matrix X. We approximate u by a vector of uncorrelated ‘residuals’, where G and are orthogonal matrices, and , while z is either 0 or a random vector uncorrelated with u satisfying , . We prove that is uncorrelated with , for any such r1, extending the results of Neudecker (1969). Building on results of Hildreth (1971) and Tiao and Guttman (1967), we show that the BAUS residual vector , where P1 is an orthonormal basis for X, minimizes each characteristic root of , while the vector rb of Theil's BLUS residuals minimizes each characteristic root of , cf. Grossman and Styan (1972). We find that if and only if the average of the singular values of is less than , and give examples to show that BAUS is often better than BLUS in this sense. |
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