The usual confidence sphere for a multivariate normal mean can be uniformly improved upon, in terms of coverage probability, by recentering it at a Stein-type estimator. However, these improved sets can have poor conditional performance. Using the theory of relevant betting procedures, which provides an objective means for assessing the conditional performance of a statistical procedure, a criterion for conditional acceptability can be established. A method of constructing such sets is outlined and applied to some recentered confidence sets. In particular, recentering at the positive-part James-Stein estimator yields a conditionally acceptable confidence set.
"Conditionally Acceptable Recentered Set Estimators." Ann. Statist. 15 (4) 1363 - 1371, December, 1987. https://doi.org/10.1214/aos/1176350598