Bootstrapping not under the null?

2025-12-11

AI Summary

This paper addresses the lack of generality and theoretical foundations in existing bootstrap hypothesis testing frameworks. We propose a unified bootstrap testing framework that accommodates both null-distribution-based resampling and diverse nonstandard bootstrap schemes. We first systematically characterize the exchangeability condition and statistical functional construction criteria, prove the local asymptotic equivalence of different resampling schemes in terms of statistical power, and identify the intrinsic mechanism behind the failure of the naive bootstrap. Leveraging empirical process theory and weak convergence analysis, we rigorously establish the asymptotic exactness and consistency of the test under fixed alternatives. An accompanying open-source R package, *BootstrapTests*, validates the theoretical properties in independence testing, linear regression coefficient testing, and copula model goodness-of-fit testing. Finite-sample simulations demonstrate that the proposed method significantly improves statistical power.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Consent frameworks and practices on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Develops a bootstrap testing framework for hypothesis tests
• Allows resampling under null and other bootstrap forms
• Applies framework to independence, regression, and goodness-of-fit tests

Innovation

• Bootstrap testing framework for hypothesis tests
• Allows resampling under null and other bootstrapping forms
• Identifies asymptotically exact and consistent test combinations