A method is given for constructing a confidence set having smallest expected measure within the class of invariant level $1 - \alpha$ confidence sets. The main assumptions are (i) that the invariance group acts transitively on the parameter space and also acts on the parametric function of interest, and (ii) that the measure satisfies a certain equivariance property. When the invariance group satisfies the conditions of the Hunt-Stein Theorem, the optimal invariant confidence set is shown to minimize the maximum expected measure among all level $1 - \alpha$ confidence sets. The method is applied in several estimation problems, including the GMANOVA problem.
"Invariant Confidence Sets with Smallest Expected Measure." Ann. Statist. 10 (4) 1283 - 1294, December, 1982. https://doi.org/10.1214/aos/1176345994