On the Conditional Consistency of the Bootstrap

Abstract

This paper focuses on the conditional consistency of the bootstrap. The target is the conditional law of a statistic given a common shock, and the bootstrap approximates it by the law of a resampled statistic conditional on the shock and the data. We call the bootstrap consistent if these laws are close in the Kolmogorov distance for almost every realization of the shock. The conditioning $σ$-algebras are allowed to be unrelated across sample sizes, and both the comparison law and the dimension of the data may vary with the sample size. We give sufficient conditions for consistency in this sense, show that they yield asymptotically correct conditional coverage for bootstrap quantiles, and provide a finite-sample bound on the coverage error. We also relate the Kolmogorov formulation to the merging of probability measures. The results are specialized to statistics depending smoothly on an estimator or on an estimated nuisance parameter.