A Special Property of Linear Estimates of the Normal Mean

Abstract

Let $X$ be a normal random variable with mean $\theta$ and variance 1 and consider the problem of estimating $\theta$ with squared error loss. If $\delta(x) = ax + b$ is a linear estimate with $0 \leqq a \leqq 1$ then it is well known that $\lambda\delta$ is an admissible proper Bayes estimate for $\lambda \in (0, 1)$. That is, all contractions of $\delta$ are proper Bayes estimates. In this note we show that no other estimates have this property.