Abstract: 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.

About the Speaker:

Yuhao Wang is an assistant professor in the Institute of Interdisciplinary Information Sciences (IIIS), Tsinghua University. Before that, Yuhao was a postdoctoral research associate at the Statistical Laboratory, which is part of the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge and a Ph.D. student from the Department of Electrical Engineering and Computer Science at Massachusetts Institute of Technology. Yuhao's main research focus is causal inference, experimental design, high dimensional statistics and distribution-free hypothesis tests.

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