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December, 1991 Constrained Minimax Estimation of the Mean of the Normal Distribution with Known Variance
Israel Feldman
Ann. Statist. 19(4): 2259-2265 (December, 1991). DOI: 10.1214/aos/1176348398


In this paper we shall discuss the estimation of the mean of a normal distribution with variance 1. The main question in this work is the existence and computation of a least favorable distribution among all the prior distributions satisfying a given set of constraints. In the following we show that if this distribution is bounded from above on some even moment, then the least favorable distribution exists and it is either normal or discrete. The support of the discrete distribution function does not have any accumulation point. The least favorable distribution is normal if and only if the second moment is bounded from above, without any other relevant constraint. These theorems shed light on the James-Stein estimator as the minimax estimator for a prior with unknown bounded variance.


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Israel Feldman. "Constrained Minimax Estimation of the Mean of the Normal Distribution with Known Variance." Ann. Statist. 19 (4) 2259 - 2265, December, 1991.


Published: December, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0745.62021
MathSciNet: MR1135176
Digital Object Identifier: 10.1214/aos/1176348398

Primary: 62F10
Secondary: 60E15 , 62C99 , 62F15

Keywords: Estimating a bounded normal mean , minimax risk , normal distribution , quadratic risk

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 4 • December, 1991
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