On Bayes and unbiased estimators of loss |
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Authors: | Dominique Fourdrinier William E Strawderman |
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Institution: | (1) Laboratoire de Mathématiques R. Salem, UMR, CNRS 6085, Université de Rouen, Mathématiques Site Colbert, 76821 Mont-Saint-Aignan, France;(2) Department of Statistics, Rutgers University, 561 Hill Center, Busch Campus, 08854-8019 Piscataway, NJ, USA |
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Abstract: | We consider estimation of loss for generalized Bayes or pseudo-Bayes estimators of a multivariate normal mean vector, θ. In
3 and higher dimensions, the MLEX is UMVUE and minimax but is inadmissible. It is dominated by the James-Stein estimator and by many others. Johnstone (1988,
On inadmissibility of some unbiased estimates of loss,Statistical Decision Theory and Related Topics, IV (eds. S. S. Gupta and J. O. Berger), Vol. 1, 361–379, Springer, New York) considered the estimation of loss for the usual
estimatorX and the James-Stein estimator. He found improvements over the Stein unbiased estimator of risk. In this paper, for a generalized
Bayes point estimator of θ, we compare generalized Bayes estimators to unbiased estimators of loss. We find, somewhat surprisingly,
that the unbiased estimator often dominates the corresponding generalized Bayes estimator of loss for priors which give minimax
estimators in the original point estimation problem. In particular, we give a class of priors for which the generalized Bayes
estimator of θ is admissible and minimax but for which the unbiased estimator of loss dominates the generalized Bayes estimator
of loss. We also give a general inadmissibility result for a generalized Bayes estimator of loss.
Research supported by NSF Grant DMS-97-04524. |
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Keywords: | Loss estimation shrinkage estimation Bayes estimation unbiased estimation superharmonicity |
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