Concentration Probabilities for Restricted and Unrestricted MLEs

We consider estimating the mean [theta] of an n dimensional normal vector X with the restriction that [theta] belongs to a closed convex set C. We investigate concentration probabilities for the restricted MLE [pi](X | C) and the MLE X. When n=2, we prove the inequality P[theta][X[set membership, variant]A+[theta]][less-than-or-equals, slant]P[theta][[pi](X | C)[set membership, variant]A+[theta]] for any [theta][set membership, variant]C and any closed convex and centrally symmetric set A. We discuss some extensions for n[greater-or-equal, slanted]3.