Improving the Reliability of Bootstrap Tests

We first propose procedures for estimating the rejection probabilities for bootstrap tests in Monte Carlo experiments without actually computing a bootstrap test for each replication. These procedures are only about twice as expensive as estimating rejection probabilities for asymptotic tersts. We then propose procedures for computing modified bootstrap P values that will often be more accurate than ordinary ones. These procedures are closely related to the double bootstrap, but they are far less computationally demanding.