Invariant prediction regions with smallest expected measure

A method is given for constructing a prediction region having smallest expected measure within the class of invariant level [beta] prediction regions. The main assumptions are that the invariance group acts transitively on the parameter space and that the measure satisfies a certain invariance property. When the invariance group satisfied the Hunt-Stein Condition, the optimal invariant prediction region minimizes the maximum expected measure among all level [beta] prediction regions. Prediction regions are constructed for: a random variable with density of arbitrary given shape but unknown location and scale; several random vectors in a multivariate regression model; and order statistics of a sample from an unspecified continuous distribution.