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.
Jiunn Tzon Hwang. George Casella. "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