Simultaneous confidence bounds for multilinear regression functions over subregions $X$ of Euclidean space are defined to be $\mu$-optimal in a class of bounds $C$, if they minimize average width with respect to $\mu$ over $X$, among all bounds in $C$ with equal coverage probability. We show that for certain simultaneous confidence bounds we can find a measure $\mu$ relative to which the bounds are $\mu$-optimal in $C$, where $C$ is a large class of bounds. Such results are obtained for bounds over finite sets, and for bounds for simple linear regression functions over finite intervals.
"Average Width Optimality of Simultaneous Confidence Bounds." Ann. Statist. 12 (4) 1199 - 1214, December, 1984. https://doi.org/10.1214/aos/1176346787