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November, 1981 Minimax Estimation of the Mean of a Normal Distribution when the Parameter Space is Restricted
P. J. Bickel
Ann. Statist. 9(6): 1301-1309 (November, 1981). DOI: 10.1214/aos/1176345646

Abstract

If $X$ is a $N(\theta, 1)$ random variable, let $\rho (m)$ be the minimax risk for estimation with quadratic loss subject to $|\theta| \leq m$. Then $\rho (m) = 1 - \pi^2/m^2 + o(m^{-2})$. We exhibit estimates which are asymptotically minimax to this order as well as approximations to the least favorable prior distributions. The approximate least favorable distributions (correct to order $m^{-2}$) have density $m^{-1} \cos^2 \big(\frac{\pi}{2m} s\big), |s| \leq m$ rather than the naively expected uniform density on $\lbrack -m, m \rbrack$. We also show how our results extend to estimation of a vector mean and give some explicit solutions.

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P. J. Bickel. "Minimax Estimation of the Mean of a Normal Distribution when the Parameter Space is Restricted." Ann. Statist. 9 (6) 1301 - 1309, November, 1981. https://doi.org/10.1214/aos/1176345646

Information

Published: November, 1981
First available in Project Euclid: 12 April 2007

zbMATH: 0484.62013
MathSciNet: MR630112
Digital Object Identifier: 10.1214/aos/1176345646

Subjects:
Primary: 62F10
Secondary: 62C99

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 6 • November, 1981
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