论文标题
回归系数测试的剩余排列测试
Residual permutation test for regression coefficient testing
论文作者
论文摘要
我们考虑测试当协变量$ p $的尺寸可以达到样本量$ n $的恒定分数时，测试单个系数是否等于线性模型中的零。在这个制度中，一个重要的主题是提出具有有限样本有效尺寸控制的测试，而无需噪声遵循强烈的分配假设。在本文中，我们提出了一种称为剩余置换测试（RPT）的新方法，该方法是通过将回归残差投影到原始设计矩阵和置换设计矩阵的柱子空间的空间正交中来构建的。只要$ p <n / 2 $，就可以证明RPT可以在固定设计下以可交换的噪声实现有限的人口尺寸有效性。此外，当真正系数至少属于$ n^{ - t/（1+t）} $的$ n^{-1+t \ in [0,1] $时，RPT对于带有界限的$（1+t）$的重尾噪声渐近强大。我们进一步证明，从最小值意义上讲，这种信号大小的要求本质上是最佳的速率。数值研究证实，RPT在具有正常和重尾噪声分布的各种模拟设置中表现良好。
We consider the problem of testing whether a single coefficient is equal to zero in linear models when the dimension of covariates $p$ can be up to a constant fraction of sample size $n$. In this regime, an important topic is to propose tests with finite-sample valid size control without requiring the noise to follow strong distributional assumptions. In this paper, we propose a new method, called residual permutation test (RPT), which is constructed by projecting the regression residuals onto the space orthogonal to the union of the column spaces of the original and permuted design matrices. RPT can be proved to achieve finite-population size validity under fixed design with just exchangeable noises, whenever $p < n / 2$. Moreover, RPT is shown to be asymptotically powerful for heavy tailed noises with bounded $(1+t)$-th order moment when the true coefficient is at least of order $n^{-t/(1+t)}$ for $t \in [0,1]$. We further proved that this signal size requirement is essentially rate-optimal in the minimax sense. Numerical studies confirm that RPT performs well in a wide range of simulation settings with normal and heavy-tailed noise distributions.
