Bayesian input in Stein estimation and a new minimax empirical Bayes estimator. (English) Zbl 0575.62010
The relationship between Stein estimation of a multivariate normal mean and Bayesian analysis is considered. The necessity to involve prior information is discussed, and the various methods of so doing are reviewed. These include direct Bayesian analyses, ad hoc utilization of prior information, restricted class Bayesian and \(\Gamma\)-minimax analyses, and type II maximum likelihood (empirical Bayes) methods.
A new empirical Bayes Stein-type estimator is developed, via the latter method, for an interesting \(\epsilon\)-contamination class of priors, and is shown to be minimax under reasonable conditions. The minimax proof contains some novel theoretical features.
A new empirical Bayes Stein-type estimator is developed, via the latter method, for an interesting \(\epsilon\)-contamination class of priors, and is shown to be minimax under reasonable conditions. The minimax proof contains some novel theoretical features.
MSC:
| 62C10 | Bayesian problems; characterization of Bayes procedures |
| 62H12 | Estimation in multivariate analysis |
| 62C12 | Empirical decision procedures; empirical Bayes procedures |
| 62F15 | Bayesian inference |
| 62C20 | Minimax procedures in statistical decision theory |
| 62P20 | Applications of statistics to economics |
Keywords:
gamma-minimax analysis; Stein estimation of a multivariate normal mean; restricted class Bayesian; type II maximum likelihood; empirical Bayes; new empirical Bayes Stein-type estimator; contamination class of priorsReferences:
| [1] | Berger, J., Admissible minimax estimation of a multivariate normal mean with arbitrary quadratic loss, Annals of Statistics, 4, 223-226 (1976) · Zbl 0322.62007 |
| [2] | Berger, J., Multivariate estimation with nonsymmetric loss functions, (Rustagi, J. S., Optimizing methods in statistics (1979), Academic Press: Academic Press New York), 5-26 · Zbl 0474.62046 |
| [3] | Berger, J., Statistical decision theory: Foundations, concepts, and methods (1980), Springer-Verlag: Springer-Verlag New York · Zbl 0444.62009 |
| [4] | Berger, J., A robust generalized Bayes estimator and confidence region for a multivariate normal mean, Annals of Statistics, 8, 716-761 (1980) · Zbl 0464.62026 |
| [5] | Berger, J., Selecting a minimax estimator of a multivariate normal mean, Annals of Statistics, 10, 81-92 (1982) · Zbl 0485.62046 |
| [6] | Berger, J., Bayesian robustness and the Stein effect, Journal of the American Statistical Association, 77, 358-368 (1982) · Zbl 0491.62030 |
| [7] | Berger, J., Estimation in continuous exponential families: Bayesian estimation subject to risk restrictions and inadmissibility results, (Gupta, S. S.; Berger, J., Statistical decision theory and related topics III (1982), Academic Press: Academic Press New York) · Zbl 0598.62002 |
| [8] | Berger, J., The Stein effect, (Kotz, S.; Johnson, N. L., Encyclopedia of statistical sciences (1983), Wiley: Wiley New York) · Zbl 0585.62002 |
| [9] | Berger, J., The robust Bayesian viewpoint, (Kadane, J., Robustness in Bayesian statistics (1983), North-Holland: North-Holland Amsterdam) |
| [10] | Berger, J., Discussion of: C. Morris, Parametric empirical Bayes inference: Theory and applications, Journal of the American Statistical Association, 78, 55-57 (1983) |
| [11] | Berger, J.; Berliner, L. M., Robust Bayes and empirical Bayes analysis with ε-contaminated priors, (Technical report no. 83-35 (1983), Statistics Department, Purdue University: Statistics Department, Purdue University West Lafayette, IN) · Zbl 0602.62004 |
| [12] | Berger, J.; Wolpert, R., The likelihood principle: A review and generalizations, Institute of Mathematical Statistics Monograph Series (1984), to appear in the · Zbl 1060.62500 |
| [13] | Berliner, L. M., A decision theoretic structure for Robust Bayesian analysis with applications to the estimation of a multivariate normal mean, (Technical report (1983), Statistics Department, Ohio State University: Statistics Department, Ohio State University Columbus, OH) · Zbl 0671.62011 |
| [14] | Bhattacharya, P. K., Estimating the mean of a multivariate normal population with general quadratic loss function, Annals of Mathematical Statistics, 37, 1819-1824 (1966) · Zbl 0151.23101 |
| [15] | Bock, M. E., Minimax estimators that shift towards a hypersphere for location vectors of spherically symmetric distributions, Journal of Multivariate Analysis (1983) · Zbl 0587.62015 |
| [16] | Box, G. E.P.; Tiao, G. C., Bayesian inference in statistical analysis (1973), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0178.22003 |
| [17] | Brown, L. D., Discussion of: J. Berger, The robust Bayesian viewpoint, (Kadane, J., Robustness in Bayesian statistics (1983), North-Holland: North-Holland Amsterdam) |
| [18] | Chen, S. Y., Restricted risk Bayes estimation, (Technical report no. 83-33 (1983), Statistics Department, Purdue University: Statistics Department, Purdue University West Lafayette, IN) |
| [19] | Efron, B.; Morris, C., Limiting the risk of Bayes and empirical Bayes estimators — Part 1: The Bayes case, Journal of the American Statistical Association, 66, 807-815 (1971) · Zbl 0229.62003 |
| [20] | Efron, B.; Morris, C., Limiting the risk of Bayes and empirical Bayes estimators — Part 2: The empirical Bayes case, Journal of the American Statistical Association, 67, 130-139 (1972) · Zbl 0231.62013 |
| [21] | Efron, B.; Morris, C., Stein’s estimation rule and its competitors — An empirical Bayes approach, Journal of the American Statistical Association, 68, 117-130 (1973) · Zbl 0275.62005 |
| [22] | Gleser, L. J., Improving inadmissible estimators under quadratic loss, (Technical report no. 83-19 (1983), Department of Statistics, Purdue University: Department of Statistics, Purdue University West Lafayette, IN) · Zbl 0592.62008 |
| [23] | Good, I. J., The estimation of probabilities (1965), M.I.T. Press: M.I.T. Press Cambridge, MA · Zbl 0168.39603 |
| [24] | Good, I. J., Some history of the hierarchical Bayesian methodology, (Bernardo, J. M.; DeGroot, M. H.; Lindley, D. V.; Smith, A. F.M., Bayesian statistics (1980), University Press: University Press Valencia) · Zbl 0467.62002 |
| [25] | Hodges, J. L.; Lehmann, E. L., The use of previous experience in reaching statistical decisions, Annals of Mathematical Statistics, 23, 392-407 (1952) · Zbl 0047.38306 |
| [26] | Hwang, J. T.; Casella, G., Minimax confidence sets for the mean of a multivariate normal distribution, Annals of Statistics, 10, 868-881 (1982) · Zbl 0508.62031 |
| [27] | James, W.; Stein, C., Estimation with quadratic loss, (Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, Vol. 1 (1961), University of California Press: University of California Press Berkeley, CA), 361-379 · Zbl 1281.62026 |
| [28] | Judge, G.; Bock, M. E., Statistical implications of pre-test and Stein rule estimators in econometrics (1978), North-Holland: North-Holland Amsterdam · Zbl 0395.62078 |
| [29] | Lindley, D. V.; Smith, A. F.M., Bayes estimates for the linear model, Journal of the Royal Statistical Society B, 34, 1-41 (1972) · Zbl 0246.62050 |
| [30] | Morris, C., Parametric empirical Bayes confidence sets, (Box, G. E.P.; Leonard, T.; Wu, C. F., Scientific inference, data analysis, and robustness (1983), Academic Press: Academic Press New York) · Zbl 0581.62033 |
| [31] | Morris, C., Parametric empirical Bayes inference: Theory and applications, Journal of the American Statistical Association, 78, 47-65 (1983) · Zbl 0506.62005 |
| [32] | Rubin, H., Robust Bayesian estimators, (Gupta, S. S.; Moore, D. S., Statistical decision theory and related topics II (1977), Academic Press: Academic Press New York) |
| [33] | Smith, G.; Campbell, F., A critique of some ridge regression methods, Journal of the American Statistical Association, 75, 74-103 (1980) · Zbl 0468.62065 |
| [34] | 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, Vol. 1 (1956), University of California Press: University of California Press Berkeley, CA), 197-206 · Zbl 0073.35602 |
| [35] | Stein, C., Estimation of the mean of a multivariate normal distribution, Annals of Statistics, 9, 1135-1151 (1981) · Zbl 0476.62035 |
| [36] | Strawderman, W. E., Proper Bayes minimax estimators of the multivariate normal mean, Annals of Mathematical Statistics, 42, 385-388 (1971) · Zbl 0222.62006 |
| [37] | Van Der Merwe, A. J.; Groenewald, P. C.N.; Nel, D. G.; Van Der Merwe, C. A., Confidence intervals for a multivariate normal mean in the case of empirical Bayes estimation using Pearson curves and normal approximations, (Technical report no. 70 (1981), Department of Mathematical Statistics, University of the Orange Free State: Department of Mathematical Statistics, University of the Orange Free State Bloemfontein) · Zbl 0675.62022 |
| [38] | Zellner, A.; Vandaele, W., Bayes-Stein estimators for \(K\)-means, regression and simultaneous equation models, (Fienberg, S.; Zellner, A., Studies in Bayesian econometrics and statistics (1971), North-Holland: North-Holland Amsterdam) |
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