Bootstrapping not under the null?

TL;DR

This paper introduces a versatile bootstrap testing framework that ensures asymptotic accuracy and consistency across various resampling schemes, with practical applications demonstrated through multiple statistical tests and an R package implementation.

Contribution

It develops a general bootstrap testing framework with conditions for asymptotic exactness and consistency, and explores the equivalence of local power functions across schemes.

Findings

Certain bootstrap schemes are asymptotically exact and consistent.

Naive bootstrap schemes may fail to perform properly.

Non-traditional bootstrap schemes can have advantages in small samples.

Abstract

We propose a bootstrap testing framework for a general class of hypothesis tests, which allows resampling under the null hypothesis as well as other forms of bootstrapping. We identify combinations of resampling schemes and bootstrap statistics for which the resulting tests are asymptotically exact and consistent against fixed alternatives. We show that in these cases the limiting local power functions are the same for the different resampling schemes. We also show that certain naive bootstrap schemes do not work. To demonstrate its versatility, we apply the framework to several examples: independence tests, tests on the coefficients in linear regression models, goodness-of-fit tests for general parametric models and for semi-parametric copula models. Simulation results confirm the asymptotic results and suggest that in smaller samples non-traditional bootstrap schemes may have advantages. This bootstrap-based hypothesis testing framework is implemented in the R package BootstrapTests.