Simultaneous prediction regions

Summary Two new methods for constructing simultaneous prediction regions are the subject of this article. Both methods simultaneously assert a collection of prediction regions, one prediction region for each future observable of interest. Both methods have the same aims: to control the overall coverage probability of the simultaneous prediction region and to keep equal the coverage probabilities of the individual prediction statements that make up the simultaneous region. The latter property is called balance.The two approaches differ in their choice of critical values. For leading cases, the first method achieves the desired overall coverage probability and the desired balance up to errors of ordern ^–1, wheren is the size of the learning sample. The second method reduces both errors to ordern ^–2. Calculating critical values in the second approach usually relies on a bootstrap algorithm.If overall coverage probability and degree of balance are instead calculatedconditionally given the learning sample, the two methods show the same asymptotic performance. This result reflects intrinsic limits on the extent to which conditional coverage probabilities can be controlled in prediction.This research was supported in part by NSF Grant DMS-87-01426. Part of the work was done while the author was a guest of Sonderforschungsbereich 123 at Universität Heidelberg