On the uniform asymptotic validity of subsampling and the bootstrap

TL;DR

This paper establishes conditions for subsampling and bootstrap methods to reliably estimate distribution quantiles and construct confidence regions and tests that are uniformly valid across a wide class of distributions, ensuring accurate inference.

Contribution

It provides new conditions under which subsampling and bootstrap methods achieve uniform asymptotic validity for a broad range of statistical procedures.

Findings

Uniform coverage probability tends to at least the nominal level.

Test sizes do not exceed the nominal level asymptotically.

Results apply to multivariate means, moment inequalities, and U-statistics.

Abstract

This paper provides conditions under which subsampling and the bootstrap can be used to construct estimators of the quantiles of the distribution of a root that behave well uniformly over a large class of distributions $\mathbf{P}$. These results are then applied (i) to construct confidence regions that behave well uniformly over $\mathbf{P}$ in the sense that the coverage probability tends to at least the nominal level uniformly over $\mathbf{P}$ and (ii) to construct tests that behave well uniformly over $\mathbf{P}$ in the sense that the size tends to no greater than the nominal level uniformly over $\mathbf{P}$. Without these stronger notions of convergence, the asymptotic approximations to the coverage probability or size may be poor, even in very large samples. Specific applications include the multivariate mean, testing moment inequalities, multiple testing, the empirical process and U-statistics.