A note on the bounded normal mean problem

The bounded normal mean problem has important applications in nonparametric function estimation. It is to estimate the mean of a normal distribution with mean restricted to a bounded interval. The minimax risk for such a problem is generally unknown. It is shown in Donoho, Liu and MacGibbon (1990) that the linear minimax risk provides a good approximation to the minimax risk. We show in this note that a better approximation can be obtained by a simple truncation of the minimax linear estimator and that the minimax linear estimator is itself inadmissible. The gain of the truncated minimax linear estimator is significant for moderate size of the mean interval, where no analytical expression for the minimax risk is available. In particular, we show that the truncated minimax linear estimator performs no more than 13% worse than the minimax estimator, comparing with 25% for the minimax linear estimator.