Minimax Estimation of the Mean of a Normal Distribution when the Parameter Space is Restricted
The Annals of Statistics · 1981 · Volume 9 · Issue 6
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.
Keywords
62C99, 62F10, estimation, Fisher information, James-Stein estimate, minimax
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