Permutation Tests at Nonparametric Rates
针对参数估计速率较慢的情形，提出一种置换检验方法，在保持有限样本精确性的同时，渐近控制检验水平，适用于两样本或单样本分割的比较，并应用于非参数函数的点估计检验。
Classical two-sample permutation tests for equality of distributions have exact size in finite samples, but they fail to control size for testing equality of parameters that summarize each distribution. This article proposes permutation tests for equality of parameters that are estimated at root-n or slower rates. Our general framework applies to both parametric and nonparametric models, with two samples or one sample split into two subsamples. Our tests have correct size asymptotically while preserving exact size in finite samples when distributions are equal. They have no loss in local asymptotic power compared to tests that use asymptotic critical values. We propose confidence sets with correct coverage in large samples that also have exact coverage in finite samples if distributions are equal up to a transformation. We apply our theory to four commonly-used hypothesis tests of nonparametric functions evaluated at a point. Lastly, simulations show good finite sample properties, and two empirical examples illustrate our tests in practice. Supplementary materials for this article are available online.