×

Minimax estimator of regression coefficient in normal distribution under balanced loss function. (English) Zbl 1232.62089

Summary: This article investigates linear minimax estimators of regression coefficient in a linear model with an assumption that the underlying distribution is a normal one with a nonnegative definite covariance matrix under a balanced loss function. Some linear minimax estimators of regression coefficients in the class of all estimators are obtained. The result shows that the linear minimax estimators are unique under some conditions.

MSC:

62H12 Estimation in multivariate analysis
62C20 Minimax procedures in statistical decision theory
62J05 Linear regression; mixed models
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alam, K., A family of admissible minimax estimators of the mean of a multivariate normal distribution, Ann. Statist., 1, 517-525 (1973) · Zbl 0259.62007
[2] M. Arashi, Problem of estimation with balanced loss function in elliptical models. Available from: <http://www.statssa.gov.za/isi2009/ScientificProgramme/IPMS/1649.pdf>; M. Arashi, Problem of estimation with balanced loss function in elliptical models. Available from: <http://www.statssa.gov.za/isi2009/ScientificProgramme/IPMS/1649.pdf>
[3] Cao, M. X., \( \Phi\) admissibility for linear estimators on regression coefficients in a general multivariate linear model under balanced loss function, J. Statist. Plann. Inference, 139, 3354-3360 (2009) · Zbl 1168.62006
[4] Efron, B.; Morris, C., Families of minimax estimators of the mean of a multivariate normal distribution, Ann. Statist., 4, 11-21 (1976) · Zbl 0322.62010
[5] Giles, J. A.; Giles, D. E.A.; Ohtani, K., The exact risks of some pre-test and Stein-type regression estimates under balanced loss, Comm. Statist. Theory Methods, 25, 2901-2924 (1996) · Zbl 0901.62086
[6] Gruber, Marvin H. J., The efficiency of shrinkage estimators with respect to Zellner’s balanced loss function, Comm. Statist. Theory Methods, 33, 235-249 (2004) · Zbl 1102.62073
[7] Hu, G. K.; Peng, P., Admissibility for linear estimators of regression coefficient in a general Gauss-Markoff model under balanced loss function, J. Statist. Plann. Inference, 140, 3365-3375 (2010) · Zbl 1207.62122
[8] Jozani, M. J.; Marchand, E.; Parsian, A., On estimation with weighted balanced-type loss function, Statist. Probab. Lett., 76, 773-780 (2006) · Zbl 1090.62007
[9] Ohtani, K.; Giles, D. E.A.; Giles, J. A., The exact risk performance of a pre-test estimator in a heteroskedastic linear regression model under the balanced loss function, Econom. Rev., 16, 119-130 (1997) · Zbl 0891.62045
[10] Ohtani, K., The exact risk of a weighted average estimator of the OLS and Stein-rule estimators in regression under balanced loss, Statist. Decisions, 16, 35-45 (1998) · Zbl 0888.62069
[11] Ohtani, K., Inadmissibility of the Stein-rule estimator under the balanced loss function, J. Econom., 88, 193-201 (1999) · Zbl 0933.62008
[12] Rao, C. R., Linear Statistical Inference and its Applications (1973), Wiley: Wiley New York · Zbl 0169.21302
[13] Rodrigues, J.; Zellner, A., Weighted balanced loss function and estimation of the mean time to failure, Comm. Statist. Theory Methods, 23, 3609-3616 (1994) · Zbl 0825.62250
[14] Wu, Q. G., A note on admissible linear estimates, Acta Math. Appl. Sinica, 5, 19-24 (1982), (in Chinese) · Zbl 0514.62012
[15] Xu, X. Z., The linear minimax estimators of regression coefficient under quadratic loss function, Ann. of Math., 14A, 5, 621-628 (1993), (in Chinese) · Zbl 0783.62048
[16] Xu, X. Z.; Wu, Q. G., Linear admissible estimators of regression coefficient under balanced loss, Acta Math. Sci., 20, 4, 468-473 (2000), (in Chinese) · Zbl 0961.62006
[17] Yu, S. H., The linear minimax estimators of estimable function in a general Gauss-Markov model under quadratic loss function, Acta Math. Appl. Sinica, 4, 26, 693-701 (2003), (in Chinese) · Zbl 1046.62005
[18] Yu, S. H., The linear minimax estimator of stochastic regression coefficients and parameters under quadratic loss function, Statist. Probab. Lett., 77, 54-62 (2007) · Zbl 1106.62009
[19] Zellner, A., Bayesian and non-Bayesian estimation using balanced loss function, (Gupta, S. S.; Berger, J. O., Statistical Decision Theory and Related Topics (1994), V. Springer: V. Springer New York), 377-390 · Zbl 0787.62035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.