## The Annals of Statistics

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

P. J. Bickel

#### 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.

#### Article information

Source
Ann. Statist., Volume 9, Number 6 (1981), 1301-1309.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176345646

Digital Object Identifier
doi:10.1214/aos/1176345646

Mathematical Reviews number (MathSciNet)
MR630112

Zentralblatt MATH identifier
0484.62013

JSTOR