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An Assumption-Free Exact Test For Fixed-Design Linear Models With Exchangeable Errors
Biometrika ( IF 2.7 ) Pub Date : 2020-09-30 , DOI: 10.1093/biomet/asaa079
Lihua Lei 1 , Peter J Bickel 2
Affiliation  

We propose the cyclic permutation test (CPT) to test general linear hypotheses for linear models. This test is non-randomized and valid in finite samples with exact type-I error $\alpha$ for arbitrary fixed design matrix and arbitrary exchangeable errors, whenever $1 / \alpha$ is an integer and $n / p \ge 1 / \alpha - 1$. The test applies the marginal rank test on $1 / \alpha$ linear statistics of the outcome vectors where the coefficient vectors are determined by solving a linear system such that the joint distribution of the linear statistics is invariant to a non-standard cyclic permutation group under the null hypothesis. The power can be further enhanced by solving a secondary non-linear travelling salesman problem, for which the genetic algorithm can find a reasonably good solution. We show that CPT has comparable power with existing tests through extensive simulation studies. When testing for a single contrast of coefficients, an exact confidence interval can be obtained by inverting the test. Furthermore, we provide a selective yet extensive literature review of the century-long efforts on this problem, highlighting the non-triviality of our test.

中文翻译:

具有可交换误差的固定设计线性模型的无假设精确检验

我们提出循环置换测试(CPT)来测试线性模型的一般线性假设。该检验是非随机的,并且在有限样本中有效,对于任意固定设计矩阵和任意可交换误差,只要 $1 / \alpha$ 是整数且 $n / p \ge 1 / \ 具有精确的 I 类误差 $\alpha$阿尔法 - 1$。该测试对结果向量的 $1 / \alpha$ 线性统计应用边际秩检验,其中系数向量通过求解线性系统来确定,使得线性统计的联合分布对于非标准循环置换组在零假设。通过求解二级非线性旅行商问题,可以进一步增强能力,遗传算法可以找到一个相当好的解决方案。我们通过广泛的模拟研究表明 CPT 具有与现有测试相当的能力。当测试单个系数的对比时,可以通过反转测试来获得准确的置信区间。此外,我们对这个问题长达一个世纪的努力进行了选择性但广泛的文献综述,突出了我们测试的非平凡性。
更新日期:2020-09-30
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