Abstract
Olman and Shmundak (1985) show that in estimating the bounded mean of a normal distribution under squared error loss the Bayes estimator with respect to the uniform prior is gammaminimax when the parameter interval is sufficiently small and the class of priors consists of all probability measures with a symmetric and unimodal density. In the present paper it is proved that this result is valid not only for the normal family but also for more general families of distributions.
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References
Berger, J.O. (1985) Statistical decision theory and Bayesian analysis, 2nd ed., Springer, Berlin-Heidelberg-New York
Eichenauer, J. (1987) Zweipunktige ungünstigste Verteilungen in Gamma-Minimax-Schätzproblemen, Dissertation, Damstadt
Eichenauer-Herrmann, J. (1989) Restricted risk Bayes estimation when the parameter interval is bounded, Preprint Nr. 1247, Fachbereich Mathematik, Technische Hochschule Darmstadt
Olman, V. and Shmundak, A. (1985) Minimax Bayes estimation of mean of normal law for the class of unimodal a priori distributions (in Russian), Proc. Acad. Sci. Estonian Physics Math. 34, 148–153
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Eichenauer-Herrmann, J. A gamma-minimax result for the class of symmetric and unimodal priors. Statistical Papers 31, 301–304 (1990). https://doi.org/10.1007/BF02924705
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DOI: https://doi.org/10.1007/BF02924705