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

If X is a N(θ, 1) random variable, let ρ (m) be the minimax risk for estimation with quadratic loss subject to |θ| ≤ m. Then ρ (m) = 1 - π²/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 cos² (π/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.