We consider the problem of testing whether a single coefficient is equal to zero in high-dimensional fixed-design linear models. In the high-dimensional setting where the dimension of covariates $p$ is allowed to be in the same order of magnitude as sample size $n$, 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 \emph{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 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:在高维固定设计线性模型中,我们考虑测试单一系数是否等于零的问题。在高维环境中,允许共差值美元与样本大小的大小相同,达到一定人口的有效性,现有方法通常要求对噪声矢量(如高山或旋转变异)进行强有力的分布假设,限制其实际应用。在本文中,我们提议了一种新的方法,即称为\emph{residal mality-demotion 测试}(RPT),该方法的构建方式是将回归残留物投射在空间或地心上,与原始和固定设计矩阵的柱体空间的结合程度相同。RPT可以证明,在固定设计下,只要有可交换的噪声,就能够实现一定的定型人口规模(如高尔西亚或旋转变异异),从而限制其实际应用。此外,RPT被证明,对于装有(1+t)美元宽度测试值的重尾声噪噪噪声音测试(RPT)-th 顺序,当真实的系数至少是Rn_xxxxxxx mess laimal depress laimal depress amin demax press aromax, press axin press airmaxxxxxxxxxxxxxyal axxxxxylxxxxxxxxxxxxxxxxxxxxxxxxxxxx