Abstract
Invariance-based randomization tests—such as permutation tests, rotation tests, or sign changes—are an important and widely used class of statistical methods. They allow drawing inferences under weak assumptions on the data distribution. Most work focuses on their type I error control properties, while their consistency properties are much less understood.
We develop a general framework to study the consistency of invariance-based randomization tests, assuming the data is drawn from a signal-plus-noise model. We allow the transforms (e.g., permutations or rotations) to be general compact topological groups, such as rotation groups, acting by linear group representations. We study test statistics with a generalized subadditivity property.
We apply our framework to a number of fundamental and highly important problems in statistics, including sparse vector detection, testing for low-rank matrices in noise, sparse detection in linear regression, and two-sample testing. Comparing with minimax lower bounds we develop, we find perhaps surprisingly that in some cases, randomization tests detect signals at the minimax optimal rate.
Funding Statement
This work was supported in part by NSF BIGDATA grant IIS 1837992 and NSF CAREER award DMS-2046874.
Acknowledgments
We thank Peter Bickel, Edward I. George, Jesse Hemerik, Ilmun Kim, Panos Toulis, and Larry Wasserman for valuable discussions. We are grateful to the Associate Editor and the referees for many helpful comments that have helped to significantly improve the paper.
Citation
Edgar Dobriban. "Consistency of invariance-based randomization tests." Ann. Statist. 50 (4) 2443 - 2466, August 2022. https://doi.org/10.1214/22-AOS2200
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