A1648
Title: Residual permutation test for high-dimensional regression coefficient testing
Authors: Kaiyue Wen - Tsinghua University (China)
Tengyao Wang - London School of Economics (United Kingdom)
Yuhao Wang - Tsinghua University and Shanghai Qi Zhi Institute (China) [presenting]
Abstract: The problem of testing whether a single coefficient is equal to zero in fixed-design linear models under a moderately high-dimensional regime is considered, 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. A new method, called residual permutation test (RPT) is proposed, 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 $t \in [0,1]$. It is further proven 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.
