**Sankhya: The Indian Journal of Statistics**

1996, Volume 58, Series A, Pt. 1, pp. 81--100

**EFFICIENCY OF LINEAR RULES FOR ESTIMATING A BOUNDED NORMAL MEAN**

By

BRANI VIDAKOVIC, *Duke University*

and

ANIRBAN DASGUPTA*, Purdue University*

SUMMARY. Inference for restricted parameters is often considerably harder than it is for the unrestricted case. A simple example is the estimation of a univariate normal mean, when the mean is known to lie in a fixed bounded interval. Levit (1980), Casella and Strawderman (1981), Bickel (1981), DasGupta (1985) describe a minimax estimator in this case. The general problem remains unsolved. An alternative to deriving the exact optimal rule is to use an easily computable rule with very good or near optimal performance. Donoho, Liu and MacGibbon (1990) demonstrate that the linear minimax rule is at most (about) 25% worse than the exact minimax rule uniformly over all bounded intervals. We consider a Bayesian version of this problem where the unknown mean is assigned a prior belonging to an appropriate family $\Gamma$ of prior distributions. Using the criterion of usual minimaxity amounts to allowing all possible prior distributions. We show that if $\Gamma$ is the class of all symmetric and unimodal priors, the linear minimax rule is at most 7.4% worse than the exact minimax rule, again uniformly over all bounded intervals. We also consider the high dimensional problem when the unknown mean is known to lie inside a sphere , and has a sperically symmetric and unimodal distribution.

*AMS (1985) subject classification.* 62C20.

*Key words and phrases.* Linear rule, gamma minimax, linear gamma minimax, Bayes risk, spherically symmetric, unimodal.