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Risk comparison of improved estimators in a linear regression model with multivariate \(t\) errors under balanced loss function. (English) Zbl 1437.62258

Summary: Under a balanced loss function, we derive the explicit formulae of the risk of the Stein-rule (SR) estimator, the positive-part Stein-rule (PSR) estimator, the feasible minimum mean squared error (FMMSE) estimator, and the adjusted feasible minimum mean squared error (AFMMSE) estimator in a linear regression model with multivariate \(t\) errors. The results show that the PSR estimator dominates the SR estimator under the balanced loss and multivariate \(t\) errors. Also, our numerical results show that these estimators dominate the ordinary least squares (OLS) estimator when the weight of precision of estimation is larger than about half, and vice versa. Furthermore, the AFMMSE estimator dominates the PSR estimator in certain occasions.

MSC:

62J05 Linear regression; mixed models
62J07 Ridge regression; shrinkage estimators (Lasso)
62H12 Estimation in multivariate analysis
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[1] Zellner, A., Bayesian and non-Bayesian analysis of the regression model with multivariate student-t error terms, Journal of the American Statistical Association, 71, 354, 400-405 (1976)
[2] Stein, C., Inadmissibility of the usual estimator for the mean of a multivariate normal distribution, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press
[3] James, W.; Stein, C., Estimation with quadratic loss, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. I, 361-379 (1961), Berkeley, Calif, USA: University of California Press, Berkeley, Calif, USA
[4] Baranchik, A. J., A family of minimax estimators of the mean of a multivariate normal distribution, Annals of Mathematical Statistics, 41, 642-645 (1970)
[5] Farebrother, R. W., The minimum mean square error linear estimator and ridge regression, Technometrics, 17, 127-128 (1975)
[6] Ohtani, K., On an adjustment of degrees of freedom in the minimum mean squared error estimator, Communications in Statistics—Theory and Methods, 25, 12, 3049-3058 (1996) · Zbl 0900.62361
[7] Giles, J. A., Pre-testing for linear restrictions in a regression model with spherically symmetric disturbances, Journal of Econometrics, 50, 3, 377-398 (1991) · Zbl 0745.62067
[8] Namba, A., PMSE performance of the biased estimators in a linear regression model when relevant regressors are omitted, Econometric Theory, 18, 5, 1086-1098 (2002) · Zbl 1033.62062
[9] Namba, A.; Ohtani, K., Risk comparison of the Stein-rule estimator in a linear regression model with omitted relevant regressors and multivariate t errors under the Pitman nearness criterion, Statistical Papers, 48, 1, 151-162 (2007) · Zbl 1134.62346
[10] Zellner, A.; Gupta, S. S.; Berger, J. O., Bayesian and non-Bayesian estimation using balanced loss functions, Statistical Decision Theory and Related Topics V, 377-390 (1994), New York, NY, USA: Springer, New York, NY, USA
[11] Giles, J. A.; Giles, D. E. A.; Ohtani, K., The exact risks of some pre-test and Stein-type regression estimators under balanced loss, Communications in Statistics—Theory and Methods, 25, 12, 2901-2924 (1996) · Zbl 0901.62086
[12] 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, Econometric Reviews, 16, 1, 119-130 (1997) · Zbl 0891.62045
[13] Ohtani, K., Inadmissibility of the Stein-rule estimator under the balanced loss function, Journal of Econometrics, 88, 1, 193-201 (1999) · Zbl 0933.62008
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