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Mean square error matrix improvements and admissibility of linear estimators. (English) Zbl 0685.62052

Summary: In the first part of this paper, the set \({\mathcal L}(Cy+c)\), comprising all linear estimators of \(\beta\) which are as good as a given unbiased estimator \(Cy+c\) with respect to the mean square error matrix criterion in at least one point of the parameter space, is investigated under the unrestricted linear regression model \(M=\{y, X\beta, \sigma^ 2I_ n\}\) and the restricted model \[ M_ 0=\{y, X\beta | R_ 0\beta =r_ 0,\sigma^ 2/I_ n\}. \] In the second part, new characterizations of the sets \({\mathcal A}\) and \({\mathcal A}_ 0\) of all linear estimators that are admissible for \(\beta\) under M and \(M_ 0\) with respect to the mean square error criterion are derived via appropriately reducing the sets \({\mathcal L}({\hat \beta})\) and \({\mathcal L}({\hat \beta}_ 0)\), where \({\hat \beta}\) and \({\hat \beta}{}_ 0\) are the minimum dispersion linear unbiased estimators of \(\beta\) in these two models. The convexity of the sets \({\mathcal L}(Cy+c)\), \({\mathcal A}\), and \({\mathcal A}_ 0\) is also pointed out.

MSC:

62J05 Linear regression; mixed models
62C15 Admissibility in statistical decision theory
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[1] Albert, A., Conditions for positive and nonnegative definiteness in terms of pseudoinverses, SIAM J. Appl. Math., 17, 434-440 (1969) · Zbl 0265.15002
[2] Baksalary, J. K.; Markiewicz, A., Admissible linear estimators in restricted linear models, Linear Algebra Appl., 70, 9-19 (1985) · Zbl 0584.62002
[3] Baksalary, J. K.; Markiewicz, A., Characterizations of admissible linear estimators is restricted linear models, J. Statist. Plann. Inference, 13, 395-398 (1986) · Zbl 0588.62014
[4] LaMotte, L. R., Admissibility in linear estimation, Ann. Statist., 10, 245-256 (1982) · Zbl 0485.62070
[5] Liski, E. P., On reduced risk estimation in linear models. Ph.D. Dissertation, Acta Univ. Tamper. Ser. A, Vol. 105 (1979)
[6] Marshall, A. W.; Olkin, I., Inequalities: Theory of Majorization and Its Applications (1979), Academic Press: Academic Press New York · Zbl 0437.26007
[7] Rao, R. C., Linear Statistical Inference and Its Applications (1965), Wiley: Wiley New York · Zbl 0137.36203
[8] Corrigendum, 7, 696 (1979) · Zbl 0421.62047
[9] Rao, C. R.; Mitra, S. K., Generalized Inverse of Matrices and Its Applications (1971), Wiley: Wiley New York
[10] Stepniak, C., Ordering of nonnegative definite matrices with application to comparison of linear models, Linear Algebra Appl., 70, 67-71 (1985) · Zbl 0578.15019
[11] Toro-Vizcarrondo, C.; Wallance, T. D., A test of the mean square error criterion for restrictions in linear regression, J. Amer. Statist. Assoc., 63, 558-572 (1968) · Zbl 0159.48001
[12] Trenkler, G., Biased Estimators in the Linear Regression Model (1981), Oelgeschlager, Gunn and Hain: Oelgeschlager, Gunn and Hain Cambridge, MA · Zbl 0471.62070
[13] Trenkler, G., Mean sqaure error matrix comparisons of estimators in linear regression, Comm. Statist. A. - Theory Methods, 14, 2495-2509 (1985) · Zbl 0594.62075
[14] Trenkler, G., Mean square error matrix comparisons among restricted least squares estimators, Sankhyã Ser. A., 49, 96-104 (1987) · Zbl 0639.62060
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