In this paper definitions are given for three types of tolerance regions. For distribution-free tolerance regions, an analytic condition is derived for the characteristic function of the region. Examples of the application of the condition are considered. For $\beta$-expectation tolerance regions, a criterion for a good tolerance region is introduced, and it is shown that the problem of finding such a tolerance region can be reduced to that of finding a good test for an equivalent hypothesis-testing problem. Best tolerance regions are obtained for a number of single variate and multivariate problems involving normal distributions.
D. A. S. Fraser. Irwin Guttman. "Tolerance Regions." Ann. Math. Statist. 27 (1) 162 - 179, March, 1956. https://doi.org/10.1214/aoms/1177728355