We consider the problem of testing whether a single coefficient is equal to zero in fixed-design linear models under a moderately high-dimensional regime, where the dimension of covariates $p$ is allowed to be in the same order of magnitude as sample size $n$. In this regime, to achieve finite-population validity, existing methods usually require strong distributional assumptions on the noise vector (such as Gaussian or rotationally invariant), which limits their applications in practice. 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.

翻译：我们考虑固定设计线性模型中单个系数是否为零的检验问题，研究场景为中等高维情形（协变量维度$p$与样本量$n$同阶）。在此框架下，现有方法为达到有限总体有效性，通常需要对噪声向量施加强分布假设（如高斯分布或旋转不变性），这限制了其实际应用。本文提出一种新方法——残差置换检验（RPT），通过将回归残差投影到原始设计矩阵与置换后设计矩阵列空间并集的正交补空间上构造检验统计量。我们证明：当$p < n / 2$时，在固定设计且噪声可交换的条件下，RPT可实现有限总体尺度有效性；此外，对于具有有界$(1+t)$阶矩的厚尾噪声（$t \in [0,1]$），当真实系数至少为$n^{-t/(1+t)}$量级时，RPT具有渐近检验功效。我们进一步证明该信号强度要求在极小极大意义下本质上是率最优的。数值实验表明，在包含正态分布与厚尾噪声的多种模拟场景中，RPT均表现优异。