Residual Permutation Test for High-Dimensional Regression Coefficient Testing

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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.