On the minimax regret estimation of a restricted normal mean, and implications

Consider estimating the mean of a normal distribution with known variance, when that mean is known to lie in a bounded interval. In a decision-theoretic framework we study finite sample properties of a class of nonlinear' estimators. These estimators are based on thresholding techniques which have become very popular in the context of wavelet estimation. Under squared errorloss we show that there exists unique minimax regret solution for the problem of selecting the threshold. For comparison, the behaviour' of linear shrinkers is also investigated. In special cases we illustrate the implications of our results for the problem of estimating the regression function in a nonparametric situation. This is possible since, as usual, a, coordinatewise application of the scalar results leads immediately to results for multivariate (sequence space) problems. Then it is well known that orthogonal transformations can be employed to turn statements about estimation over coefficient bodies in sequence space into statements about estimation over classes of smooth functions in noisy data. The performance of the proposed minimax regret optimal curve estimator is demonstrated by simulated data examples.