Comparison of regression estimators using Pitman measures of nearness. (English) Zbl 1099.62525

Summary: Using the criterion of Pitman measure of nearness, the paper provides a theoretical result about the performance of any two members of a class of ridge-type estimators of the regression parameter. Based on the result, it is possible to establish conditions under which a member of the ridge-type class of estimators is better than the standard least squares estimator in the Pitman-nearness sense.


62J07 Ridge regression; shrinkage estimators (Lasso)
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[1] Bertuzzi, A.; Gandolfi, A., Ridge regression versus OLS by Pitman’s closeness under quadratic and Fisher’s loss, Communications in Statistics — Theory and Methods, 20, 3581-3590 (1991) · Zbl 0800.62414
[2] Conerly, M. D.; Hardin, M. J., A comparison of biased regression estimators using a Pitman Nearness criterion, Journal of Statistical Computation and Simulations, 34, 75-95 (1990) · Zbl 0726.62117
[3] Hocking, R. R.; Speed, F. M.; Lynn, M. J., A class of biased estimators in linear regression, Technometrics, 18, 425-437 (1976) · Zbl 0346.62046
[4] Hoerl, A. E.; Kennard, R. W., Ridge regression: Biased estimation for non-orthogonal problems, Technometrics, 12, 55-67 (1970) · Zbl 0202.17205
[5] Hoerl, A. E.; Kennard, R. W., Ridge regression: Applications to non-orthogonal problems, Technometrics, 12, 69-82 (1970) · Zbl 0202.17206
[6] Keating, J.P., Mason, R.L. and Sen, P.K.(1993): Pitman Measure of Closeness: A Comparison of Statistical Estimators. Society for Industrial and Applied Mathematics (SIAM), (monograph). · Zbl 0779.62019
[7] Marquardt, D. W., Generalized inverses, ridge regression, biased linear estimation and non-linear estimation, Technometrics, 12, 591-612 (1970) · Zbl 0205.46102
[8] Mason, R. L.; Blaylock, N. W., Ridge regression estimator comparisons using Pitman’s Measure of Closeness, Communications in Statistics-Theory and Methods, 20, 3629-3641 (1991)
[9] Mason, R. L.; Keating, J. P.; Sen, P. K.; Blaylock, N. W., Comparison of linear estimators using Pitman’s Measure of Closeness, Journal of American Statistical Association, 85, 579-581 (1990) · Zbl 0702.62049
[10] Mayer, L. S.; Willke, T. A., On biased estimation in linear models, Technometrics, 15, 497-508 (1973) · Zbl 0265.62017
[11] Pitman, E. J.G., The closest estimates of statistical parameters, Proceedings of the Cambridge Philosophical Society, 33, 212-222 (1937) · JFM 63.0515.03
[12] Obenchain, R. L., Ridge analysis following a preliminary test of shrunken hypothesis, Technometrics, 17, 431-445 (1975) · Zbl 0324.62050
[13] Obenchain, R. L., Good and optimal ridge estimators, Annals of Statistics, 6, 1111-1121 (1978) · Zbl 0384.62059
[14] Rao, C. R.; Csorgo, M.; Dawson, D. A.; Rao, J. N.K.; Saleh, A. A. M. E., Some comments on the minimum mean square error as a criterion of estimation, Statistics and Related Topics, 123-143 (1981), Amsterdam: North-Holland, Amsterdam
[15] Trenkler, D., Verallgemeinerte Ridge Regression, Mathematical Systems in Economics 104 (1986), Frankfurt/Main: Anton Hain Verlag, Frankfurt/Main
[16] Trenkler, G., Biased estimators in linear regression model, Mathematical Systems in Economics 58. Oelgeschlager (1981), Cambridge, Massachussets: Gunn/Hain, Cambridge, Massachussets · Zbl 0471.62070
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