Abstract
In this paper, we consider a linear regression model with multivariate t error terms and derive the explicit formula of the mean squared error (MSE) of the two-stage hierarchial information (2SHI) estimator. It is shown by numerical evaluations that the 2SHI estimator has smaller MSE than the positive-part Stein-rule (PSR) estimator over a wide region of the parameter space.
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References
Baranchik, A.J. (1970), A family of minimax estimators of the mean of a multivariate normal distribution, Annals of Mathematical Statistics, 41, 642–645.
Blattberg, R.C. and N.J. Gonedes (1974), A comparison of the stable and Student distributions as statistical models for stock prices, Journal of Business, 47, 244–280.
Fama, E.F. (1965), The behavior of stock market prices, Journal of Business, 38, 34–105.
Farebrother, R.W. (1975), The minimum mean square error linear estimator and ridge regression. Technometrics, 17, 127–128.
Giles, J.A. (1991), Pre-testing for linear restrictions in a regression model with spherically symmetric disturbances, Journal of Econometrics, 50, 377–398.
Giles, J.A. (1992), Estimation of the error variance after a preliminary test of homogeneity in a regression model with spherically symmetric disturbances, Journal of Econometrics, 53, 345–361.
James, W. and C. Stein (1961), Estimation with quadratic loss, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability 1 (Berkeley, University of California Press), 361–379.
Judge, G., S. Miyazaki and T. Yancey (1985), Minimax estimators for the location vectors of spherically symmetric densities, Econometric Theory, 1, 409–417.
Ohtani, K. (1991), Small sample properties of the interval constrained least squares estimator when error terms have a multivariate t distribution, Journal of the Japan Statistical Society, 21, 197–204.
Ohtani, K. (1993), Testing for equality of error variances between two linear regressions with independent multivariate t errors, Journal of Quantitative Economics, 9, 85–97.
Ohtani, K. (1996), On an adjustment of degrees of freedom in the minimum mean squared error estimator, Communications in Statistics-Theory and Methods, 25, 3049–3058.
Ohtani, K. and H. Hasegawa (1993), On small sample properties of R 2 in a linear regression model with multivariate t errors and proxy variables, Econometric Theory, 9, 504–515.
Ohtani, K. and J. Giles (1993), Testing linear restrictions on coefficients in a linear regression model with proxy variables and spherically symmetric disturbances, Journal of Econometrics, 57, 393–406.
Prucha, I.R. and H. Kelejian (1984), The structure of simultaneous equation estimators: a generalization towards nonnormal distributions, Econometrica, 52, 721–736.
Singh, R.S. (1988), Estimation of error variance in linear regression models with error having multivariate Student-t distribution with unknown degrees of freedom, Economics Letters, 27, 47–53.
Singh, R.S. (1991), James-Stein rule estimators in linear regression models with multivariate-t distributed error, Australian Journal of Statistics, 33, 145–158.
Stein, C. (1956), Inadmissibility of the usual estimator for the mean of a multivariate normal distribution, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, University of California Press), 197–206.
Sutradhar, B.C. (1988), Testing linear hypothesis with t error variable, Sankhya-B, 50, 175–180.
Sutradhar, B.C. and M.M. Ali (1986), Estimation of the parameters of a regression model with a multivariate t error variable, Communications in Statistics-Theory and Methods, 15, 429–450.
Tran Van Hoa (1985), The inadmissibility of the Stein estimator in normal multiple regression equations, Economics Letters, 19, 39–42.
Tran Van Hoa (1992), Modelling Output Growth, Economics Letters, 38, 279–284.
Tran Van Hoa (1993), The mixture properties of the 2SHI estimators in linear regression models, Statistics and Probability Letters, 16, 111–115.
Tran Van Hoa and A. Chaturvedi (1990), Further results on the two-stage hierarchial information (2SHI) estimators in the linear regression models, Communications in Statistics-Theory and Methods, 19, 4697–4704.
Tran Van Hoa and A. Chaturvedi (1993), Asymptotic approximations to the gain of the 2SHI over Stein estimators in linear regression models when the disturbances are small, Communications in Statistics-Theory and Methods, 22, 2777–2782.
Tran Van Hoa and A. Chaturvedi (1997), Performance of the 2SHI estimator under the generalised Pitman nearness criterion, Communications in Statistics-Theory and Methods, 26, 1227–1238.
Ullah, A. and V. Zinde-Walsh (1984), On the robustness of LM, LR, and Wald tests in regression models, Econometrica, 52, 1055–1066.
Zellner, A. (1976), Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error terms, Journal of the American Statistical Association, 71, 400–405.
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Namba, A. MSE performance of the 2SHI estimator in a regression model with multivariate t error terms. Statistical Papers 42, 81–96 (2001). https://doi.org/10.1007/s003620000041
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DOI: https://doi.org/10.1007/s003620000041