For the problem of estimating a $p$-variate normal mean, the existence of confidence procedures which dominate the usual one, a sphere centered at the observations, has long been known. However, no explicit procedure has yet been shown to dominate. For $p \geq 4$, we prove that if the usual confidence sphere is recentered at the positive-part James Stein estimator, then the resulting confidence set has uniformly higher coverage probability, and hence is a minimax confidence set. Moreover, the increase in coverage probability can be quite substantial. Numerical evidence is presented to support this claim.
"Minimax Confidence Sets for the Mean of a Multivariate Normal Distribution." Ann. Statist. 10 (3) 868 - 881, September, 1982. https://doi.org/10.1214/aos/1176345877