Concentration Probabilities for Restricted and Unrestricted MLEs

We consider estimating the mean θ of an n dimensional normal vector X with the restriction that θ belongs to a closed convex set C. We investigate concentration probabilities for the restricted MLE π(X ∣ C) and the MLE X. When n=2, we prove the inequality Pθ[X∈A+θ]⩽Pθ[π(X ∣ C)∈A+θ] for any θ∈C and any closed convex and centrally symmetric set A. We discuss some extensions for n⩾3.