Minimax optimality of permutation tests
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置换检验的极小极大最优性
- 研究问题
- 在原假设分布具有相应置换不变性时，如何分析置换检验的非渐近功效，以及两样本和独立性检验中的极小极大表现？
- 方法与思路
- 作者提出非渐近分析框架，以 U 统计量构造置换检验，通过新的耦合思路建立指数集中界，并构造适应未知光滑度的检验。
- 材料陈述的贡献
- 摘要称分析了离散及连续情形的极小极大表现，建立置换 U 统计量集中界，并将框架用于加权统计量和高斯核统计量等例子。
- 阅读关注点
- 应核对置换不变性、备择集合和分离尺度的定义，确认各场景的最优性结论，并检查适应未知光滑度所付出的功效代价。
- 阅读边界
- 依据版本未注明的 OpenAlex 索引摘要，具体检验条件与最优界未提供；这些置换功效结论不能直接解释为因果随机化推断结果。
原文摘要
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.