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Concentration Probabilities for Restricted and Unrestricted MLEs

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Abstract

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](XÂ |Â 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](XÂ |Â 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.

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