Consistency of invariance-based randomization tests

TL;DR

This paper develops a mathematical framework using representation theory to analyze the consistency of invariance-based randomization tests, demonstrating their effectiveness in various statistical signal detection problems.

Contribution

It introduces a general framework for the consistency analysis of invariance-based tests using representation theory, applicable to broad classes of transformations and statistical models.

Findings

Randomization tests can be consistent in signal detection.

Some tests achieve minimax optimal detection rates.

The framework applies to diverse problems like sparse vectors and low-rank matrices.

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 and a set of results on the consistency of invariance-based randomization tests in signal-plus-noise models. Our framework is grounded in the deep mathematical area of representation theory. We allow the transforms to be general compact topological groups, such as rotation groups, acting by general linear group representations. We study test statistics with a generalized sub-additivity 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 find perhaps surprisingly that in some cases, randomization tests detect signals at the minimax optimal rate.