Abstract
In this paper, we study the issue of admissibility of linear estimated functions of parameters in the multivariate linear model with respect to inequality constraints under a matrix loss and a matrix balanced loss. Under the matrix loss, when the model is not constrained, the results in the class of non-homogeneous linear estimators [Xie, 1989, Chinese Sci. Bull., 1148–1149; Xie, 1993, J. Multivariate Anal., 1071–1074] showed that the admissibility under the matrix loss and the trace loss is equivalent. However, when the model is constrained by the inequality constraints, we find this equivalency is not tenable, our result shows that the admissibility of linear estimator does not depend on the constraints again under this matrix loss, but it is contrary under the trace loss [Wu, 2008, Linear Algebra Appl., 2040–2048], and it is also relative to the constraints under another matrix loss [He, 2009, Linear Algebra Appl., 241–250]. Under the matrix balanced loss, the necessary and sufficient conditions that the linear estimators are admissible in the class of homogeneous and non-homogeneous linear estimators are obtained, respectively. These results will support the theory of admissibility on the linear model with inequality constraints.
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References
Baksalary, J.K., Markiewicz, A. Admissible linear estimators in the general Gauss-Markov model. Journal of statistical planning and Inference, 19: 349–359 (1988)
Cao, M. F admissibility for linear estimators on regression coefficients in a general multivariate linear model under balanced loss function. Journal of statistical planning and Inference, 139: 3354–3360 (2009)
Cohen, A All admissibility estimates of the mean vector. Ann. Math. Statist., 37: 458–463 (1966)
Deng, Q.R., Chen, J.B. Admissibility of nonhomogeneous linear estimators in linear model with respect to an incomplete ellipsoidal restriction under matrix loss function. Chinese Ann. Math., 18(A): 33–40 (1997)
Dong, L., Wu, Q. The sufficient and necessary conditions of admissible linear estimates for random regression coefficients and parameters under the quadratic loss function. Acta Math. Sinica, 31: 145–157 (1988)
He, D.J., Guo, D.W. Admissibility of linear estimators with respect to inequality constraints under matrix loss function. Linear Algebra Appl., 430: 241–250 (2009)
Hoffmann, K. admissibility of linear estimation with respect to restricted parameter sets. Math. Oper. Statist. Ser. Statist., 8: 425–438 (1977)
LaMotte, L.R. Admissibility in linear estimation. Ann. Math. Statist., 10: 245–255 (1982)
LaMotte, L.R. On limits of uniquely best linear estimators. Metrika, 45: 197–211 (1997)
Lu, C.Y. Admissibility of inhomogeneous linear estimators in linear model with respect to an incomplete ellipsoidal restriction. Commun. Statist. Theory and Methods, 24: 1737–1742 (1995)
Lu, C.Y., Shi, N.Z. Admissible linear estimators in linear models with respect to inequality constrains. Linear Algebra Appl., 354: 187–194 (2002)
Rao, C.R. Estimation of parameter in a linear models. Ann. Math. Statist., 4: 1023–1037 (1976)
Stein, C. Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc. Third Berkeley Symp. Math. Statist. Probab., 1: 197–206 (1956)
Stepniak, C. A complete class for linear estimation in a general linear model. Ann. Inst. Statist. Math. A, 39: 563–573 (1987)
Stepniak, C. Admissible linear estimators in mixed linear models. J. Multivariate Anal., 31: 90–106 (1989)
Synowka-Bejenka, E., Zontek, S. A characterization of admissible linear estimators of a fixed and random effects in linear models. Metrika, 68: 157–172 (2008)
Wu, Q.G. Admissibility of linear estimators of regression coefficient in a general Gauss-Markoff model. Acta Math. Appl. Sinica, 9: 251–256 (1986)
Wu, J.H. Admissibility of linear estimators in multivariate linear models with respect to inequality constraints. Linear Algebra Appl, 428: 2040–2048 (2008)
Xie, M. Admissibility for linear estimates on regression coefficients. Chinese Sci. Bull., 34: 1448–1449 (1989)
Xie, M. all admissible estimates of the mean matrix. J. Multivariate Anal., 35: 1071–1074 (1993)
Xu, X., Wu, Q. Linear admissible estimators of regression coefficient under balanced loss. Acta Math. Sci., 20: 468–473 (2000) (in Chinese)
Zhang, S.L., Gui, W.H. Admissibility of linear estimators in a growth curve model subject to an incomplete ellipsoidal restriction. Acta Mathematica Scientia, 28: 194–200 (2008)
Zhang, S.L., Gui, W.H., Liu, G. Characterization of admissible linear estimators in the general growth curve model with respect to an incomplete ellipsoidal restriction. Linear Algebra Appl., 431: 120–131 (2009)
Zhang, S.L., Liu, G., Gui, W.H. Admissibility in the general multivariate linear model with respect to restricted parameter set. Journal of Inequalities and Applications, 2009: ID 718927 (2009)
Zhu, X.H., Lu, C.Y. Admissibility of in-homegeneous linear estimators of regression coefficient. Chinese Journal Applied Probability and Statistics, 2: 97–99 (1986)
Zontek, S. Admissibility of limits of the unique locally best linear estimators with application to variance components models. Probab. Math. Statist., 9: 29–44 (1988)
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The authors greatly appreciate helpful suggestions of the referees and the editor that greatly improved the paper.
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Supported in part by the National Natural Science Foundation of China under Grant No. 61070236 and 11271147.
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Zhang, Sl., Qin, H. & Wu, Cc. Admissibility of linear estimators with respect to inequality constraints under some loss functions. Acta Math. Appl. Sin. Engl. Ser. 33, 1073–1082 (2017). https://doi.org/10.1007/s10255-017-0719-5
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DOI: https://doi.org/10.1007/s10255-017-0719-5