Improved Confidence Statements for the Usual Multivariate Normal Confidence Set
The usual multivariate normal confidence set has reported confidence 1—α, which is equal to its coverage probability. If we take a decision theoretic view, and attempt to estimate the coverage, we find that 1— α is an inadmissible estimator in more than four dimensions. We establish this fact and, moreover, exhibit adaptive confidence estimators that appear to dominate 1— α. These new confidence estimators are developed through empirical Bayes arguments and approximations. They allow us to attach confidence that is uniformly greater than 1— α. We provide necessary conditions, and strong numerical evidence to support our domination claims.
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