Minimax Estimation of the Mean of a Normal Distribution when the Parameter Space is Restricted

If X is a N (, 1) random variable, let (m) be the minimax risk for estimation with quadratic loss subject to || m. Then (m) = 1 - ²/m² + o (m^-2). We exhibit estimates which are asymptotically minimax to this order as well as approximations to the least favorable prior distributions. The approximate least favorable distributions (correct to order m^-2) have density m^-1 ² (2m s), |s| m rather than the naively expected uniform density on -m, m. We also show how our results extend to estimation of a vector mean and give some explicit solutions.