The size of the bootstrap test of hypotheses is studied for the normal and exponential one and two-sample problems. It is found that the size depends not only on the problem, but on the choice of test statistic and the nominal level. In some special cases, the bootstrap test is UMP, but in other cases, it can be totally useless, such as being completely randomized or rejecting the null hypothesis with probability one. More importantly, the size is usually greater than the nominal level, even in the limit as the sample size goes to infinity.
Digital Object Identifier: 10.1214/09-LNMS5708