MINIMAX OPTIMALITY OF PERMUTATION TESTS

• 成果类型: Article
• 署名作者: Kim, Ilmun; Balakrishnan, Sivaraman; Wasserman, Larry
• 署名单位: Yonsei University; Carnegie Mellon University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/21-AOS2103
• 发表日期: 2022
• 页码: 225-251
• 关键词: high-dimensional multinomials; large-sample power; asymptotic expansions; randomization tests; MULTIVARIATE; bootstrap; hypotheses; dependence; density

摘要

Permutation tests are widely used in statistics, providing a finite-sample guarantee on the type I error rate whenever the distribution of the samples under the null hypothesis is invariant to some rearrangement. Despite its increasing popularity and empirical success, theoretical properties of the permutation test, especially its power, have not been fully explored beyond simple cases. In this paper, we attempt to partly fill this gap by presenting a general nonasymptotic framework for analyzing the minimax power of the permutation test. The utility of our proposed framework is illustrated in the context of two-sample and independence testing under both discrete and continuous settings. In each setting, we introduce permutation tests based on U-statistics and study their minimax performance. We also develop exponential concentration bounds for permuted U-statistics based on a novel coupling idea, which may be of independent interest. Building on these exponential bounds, we introduce permutation tests, which are adaptive to unknown smoothness parameters without losing much power. The proposed framework is further illustrated using more sophisticated test statistics including weighted U-statistics for multinomial testing and Gaussian kernel-based statistics for density testing. Finally, we provide some simulation results that further justify the permutation approach.