Abstract

We consider the problem of testing whether a single coefficient is equal to zero in fixed-design linear models with moderately high-dimensional covariates. In the moderate 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 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 minimax rate optimal. Numerical studies confirm that RPT performs well in a wide range of simulation settings with normal and heavy-tailed noise distributions. This is based on joint works with Kaiyue Wen and Tengyao Wang.