Bootstrap Sample Size in Nonregular Cases

We study the bootstrap estimator of the sampling distribution of a given statistic in some nonregular cases where the given statistic is nonsmooth or not-so-smooth. It is found that the ordinary bootstrap, based on a bootstrap sample of the same size as the original data set, produces an inconsistent bootstrap estimator. On the other hand, when we draw a bootstrap sample of a smaller size with the ratio of the size of the bootstrap sample over the size of the original data set tending to zero, the resulting bootstrap estimator is consistent. Examples of these nonregular cases are given, including the cases of functions of means with null first-order derivatives, differentiable statistical functionals with null influence function, nondifferentiable functions of means, and estimators based on some test results.