Estimation of the Mean of a Univariate Normal Distribution with Known Variance

We consider the estimation of the unknown mean "e" of a univariate normal distribution N("e", 1) given a single observation "x". We wish to obtain an estimator which is admissible and has good risk (and regret) properties. We first argue that the "usual" estimator "t" ("x") e "x" is not necessarily suitable. Next, we show that the traditional pretest estimator of the mean has many undesirable properties. Thus motivated, we suggest the Laplace estimator, based on a "neutral" prior for "e", and obtain its properties. Finally, we compare the Laplace estimator with a large class of (inadmissible) estimators and show that the risk properties of the Laplace estimator are close to those of the minimax regret estimator from this large class. Thus, the Laplace estimator has good risk (regret) properties as well. Questions of admissibility, risk and regret are reviewed in the appendix. Copyright Royal Economic Society 2002