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Choice of hierarchical priors: Admissibility in estimation of normal means. (English) Zbl 0865.62004
Summary: In hierarchical Bayesian modeling of normal means, it is common to complete the prior specification by choosing a constant prior density for unmodeled hyperparameters (e.g., variances and highest-level means). This common practice often results in an inadequate overall prior, inadequate in the sense that estimators resulting from its use can be inadmissible under quadratic loss. In this paper, hierarchical priors for normal means are categorized in terms of admissibility and inadmissibility of resulting estimators for a quite general scenario. The Jeffreys prior for the hypervariance and a shrinkage prior for the hypermeans are recommended as admissible alternatives. Incidental to this analysis is presentation of the conditions under which the (generally improper) priors result in proper posteriors.

MSC:
62C15Statistical admissibility
62F15Bayesian inference
62H12Multivariate estimation
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References:
[1] Andrews, R., Berger, J. and Smith, M. (1993). Bayesian estimation of fuel economy potential due to technology improvements. Case Studies in Bayesian Statistics. Lecture Notes in Statist. 83. Springer, New York.
[2] Angers, J. F. (1987). Development of robust Bay es estimators for a multivariate normal mean. Ph.D. dissertation, Dept. Statistics, Purdue Univ.
[3] Baranchik, A. (1964). Multiple regression and estimation of the mean of a multivariate normal distribution. Technical Report 51, Dept. Statistics, Stanford Univ.
[4] Berger, J. (1976). Admissible minimax estimation of a multivariate normal mean under arbitrary quadratic loss. Ann. Statist. 4 223-226. · Zbl 0322.62007 · doi:10.1214/aos/1176343356
[5] Berger, J. (1980). A robust generalized Bay es estimator and confidence region for a multivariate normal mean. Ann. Statist. 8 716-761. · Zbl 0464.62026 · doi:10.1214/aos/1176345068
[6] Berger, J. (1984). The robust Bayesian viewpoint. In Robustness in Bayesian Statistics (J. Kadane, ed.). North-Holland, Amsterdam.
[7] Berger, J. (1985). Statistical Decision Theory and Bayesian Analy sis, 2nd ed. Springer, New York. Berger, J. and Bernardo, J. (1992a). On the development of the reference prior method. In Bayesian Statistics (J. M. Bernardo, J. O. Berger A. P. Dawid and A. F. M. Smith, eds.) 4 35-60. Oxford Univ. Press. Berger, J. and Bernardo, J. (1992b). Reference priors in a variance components problem. Bayesian Analy sis in Statistics and Econometrics. Lecture Notes in Statist. 75 177- 194. · Zbl 0572.62008
[8] Berger, J. and Deely, J. (1988). A Bayesian approach to ranking and selection of related means with alternatives to AOV methodology. J. Amer. Statist. Assoc. 83 364-373. JSTOR: · Zbl 0672.62041 · doi:10.2307/2288851 · http://links.jstor.org/sici?sici=0162-1459%28198806%2983%3A402%3C364%3AABATRA%3E2.0.CO%3B2-K&origin=euclid
[9] Berger, J. and Robert, C. (1990). Subjective hierarchical Bay es estimation of a multivariate normal mean: on the frequentist interface. Ann. Statist. 18 617-651. · Zbl 0719.62043 · doi:10.1214/aos/1176347619
[10] Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analy sis. Addison-Wesley, Reading, MA. · Zbl 0271.62044
[11] Brown, L. (1971). Admissible estimators, recurrent diffusions, and insoluble boundary value problems. Ann. Math. Statist. 42 855-903. · Zbl 0246.62016 · doi:10.1214/aoms/1177693318
[12] Ghosh, M. (1992). Hierarchical and empirical Bay es multivariate estimation. In Current Issues in Statistical Inference. IMS, Hay ward, CA. · Zbl 0798.62014 · doi:10.1214/lnms/1215458844
[13] Goel, P. K. (1983). Information measures and Bayesian hierarchical models. J. Amer. Statist. Assoc. 78 408-410. JSTOR: · Zbl 0528.62005 · doi:10.2307/2288648 · http://links.jstor.org/sici?sici=0162-1459%28198306%2978%3A382%3C408%3AIMABHM%3E2.0.CO%3B2-A&origin=euclid
[14] Good, I. J. (1983). The robustness of a hierarchical model for multinomials and contingency tables. In Scientific Inference, Data Analy sis, and Robustness (G. E. P. Box, T. Leonard and C. F. Wu, eds.). Academic Press, New York. · Zbl 0572.62046
[15] Hill, B. (1977). Exact and approximate Bayesian solutions for inference about variance components and multivariate inadmissibility. In New Development in the Application of Bayesian Methods (A. Ay kac and C. Brumet, eds.). North-Holland, Amsterdam.
[16] Jeffrey s, H. (1961). Theory of Probability. Oxford Univ. Press.
[17] Kelker, D. (1970). Distribution theory of spherical distributions and a location-scale parameter. Sankhy?a Ser. A 32 419-430. · Zbl 0223.60008
[18] Morris, C. (1983). Parametric empirical Bay es inference: theory and applications (with discussion). J. Amer. Statist. Assoc. 78 47-65. JSTOR: · Zbl 0506.62005 · doi:10.2307/2287098 · http://links.jstor.org/sici?sici=0162-1459%28198303%2978%3A381%3C47%3APEBITA%3E2.0.CO%3B2-S&origin=euclid
[19] Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc. Third Berkeley Sy mp. Math. Statist. Probab. 1 197-206. Univ. California Press, Berkeley. · Zbl 0073.35602
[20] Strawderman, W. (1971). Proper Bay es minimax estimators of the multivariate normal mean. Ann. Math. Statist. 42 385-388. · Zbl 0222.62006 · doi:10.1214/aoms/1177693528
[21] Strawderman, W. (1973). Proper Bay es minimax estimators of the multivariate normal mean for the case of common unknown variances. Ann. Statist. 1 1189-1194. · Zbl 0286.62007 · doi:10.1214/aos/1176342567
[22] Ye, K. (1993). Sensitivity study of the reference priors in a random effect model. Technical report, Dept. Statistics, Virginia Poly technic Institute and State Univ., Blacksburg.
[23] Zheng, Z. (1982). A class of generalized Bay es minimax estimators. In Statistical Decision Theory and Related Topics (S. S. Gupta and J. Berger, eds.) 3. Academic Press, New York. · Zbl 0585.62017