Consistency of the reduced bootstrap for sample means

In this paper we consider the reduced bootstrap, that consists of only considering those bootstrap samples satisfying k1⩽νn⩽k2, for some 1⩽k1⩽k2⩽n, where νn is the number of distinct original observations in the bootstrap sample. We show that, under mild conditions on k1 and k2, the reduced bootstrap estimator of the distribution of the sample mean is consistent. We also give conditions for it to be second order accurate and extend the obtained results to statistics that can be expressed as a smooth function of a multivariate sample mean and to their studentized versions.