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Minimax Estimation of the Mean of a Normal Distribution when the Parameter Space is Restricted

P. J. Bickel
The Annals of Statistics
Vol. 9, No. 6 (Nov., 1981), pp. 1301-1309
Stable URL: http://www.jstor.org/stable/2240419
Page Count: 9
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Minimax Estimation of the Mean of a Normal Distribution when the Parameter Space is Restricted
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Abstract

If X is a N(θ, 1) random variable, let ρ (m) be the minimax risk for estimation with quadratic loss subject to |θ| ≤ m. Then ρ (m) = 1 - π2/m2 + 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 [ -m, m ]. We also show how our results extend to estimation of a vector mean and give some explicit solutions.

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