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Bounds for the trace of the difference of the covariance matrices of the OLSE and BLUE. (English) Zbl 0753.62033
Summary: In the linear model \(\{y,X\beta,V\}\), the inefficiency of the ordinary least squares estimator of \(X\beta\) can be measured as the difference of the covariance matrices of the ordinary least squares estimator and the best linear unbiased estimator of \(X\beta\). C. R. Rao [ibid. 70, 249-255 (1985; Zbl 0594.62073)] gave an upper bound for the trace of this difference, assuming that the covariance matrix \(V\) is positive definite. In this paper, we generalize this result to the situation when \(V\) is allowed to be singular.

62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models
Full Text: DOI
[1] Baksalary, J.K.; Puntanen, S., Generalized matrix versions of the Cauchy-Schwarz and Kantorovich inequalities, Aequationes math., 41, 103-110, (1991) · Zbl 0723.15017
[2] Bartmann, F.C.; Bloomfield, P., Inefficiency and correlation, Biometrika, 68, 67-71, (1981) · Zbl 0472.62073
[3] Bloomfield, P.; Watson, G.S., The inefficiency of least squares, Biometrika, 62, 121-128, (1975) · Zbl 0308.62056
[4] Knott, M., On the minimum efficiency of least squares, Biometrika, 62, 129-132, (1975) · Zbl 0308.62057
[5] Marsaglia, G.; Styan, G.P.H., Rank conditions for generalized inverses of partitioned matrices, Sankhyā ser. A, 36, 437-442, (1974) · Zbl 0309.15002
[6] Marshall, A.W.; Olkin, I., Inequalities: theory of majorization and its applications, (1979), Academic New York · Zbl 0437.26007
[7] Puntanen, S., Properties of the canonical correlations between the least squares fitted values and the residuals, (), 269-284
[8] Puntanen, S., Properties of the covariance matrix of the BLUE in the general linear model, (), 425-430
[9] Puntanen, S., On the relative goodness of ordinary least squares estimation in the general linear model, Ph.D. dissertation, 216, (1987), Acta Univ. Tamper. Ser. A
[10] Puntanen, S.; Styan, G.P.H., The equality of the ordinary least squares estimator and the best linear unbiased estimator, Amer. statist., 43, 151-161, (1989), [commented by O. Kempthorne on pp. 161-162 and by S.R. Searle on pp. 162-163; reply by the authors on p.164].
[11] Rao, C.R., Least squares theory using an estimated dispersion matrix and its application to measurement of signals, (), 355-372 · Zbl 0189.18503
[12] Rao, C.R., The inefficiency of least squares: extensions of the Kantorovich inequality, Linear algebra appl., 70, 249-255, (1985) · Zbl 0594.62073
[13] Styan, G.P.H., On some inequalities associated with ordinary least squares and the Kantorovich inequality, Festschrift for eino haikala on his seventieth birthday, acta univ. tamper. ser. A, 153, 158-166, (1983)
[14] von Neumann, J., Some matrix-inequalities and metrization of matric-space, (), 1, 205-219, (1937), reprinted in · Zbl 0017.09802
[15] Watson, G.S., Serial correlation in regression analysis I, Biometrika, 42, 327-341, (1955) · Zbl 0068.33201
[16] Zyskind, G., On canonical forms, non-negative covariance matrices and best and simple least squares linear estimators in linear models, Ann. math. statist., 38, 1092-1109, (1967) · Zbl 0171.17103
[17] Zyskind, G.; Martin, F.B., On best linear estimation and a general Gauss-Markov theorem in linear models with arbitrary nonnegative covariance structure, SIAM J. appl. math., 17, 1190-1202, (1969) · Zbl 0193.47301
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