Maximum-type tests for high-dimensional regression coefficients using Wilcoxon scores

Abstract

In this article, we develop new maximum-type tests to infer the overall significance of coefficients in high-dimensional linear models based on the Wilcoxon scores. The proposed testing procedures are free of error variance estimation and robust to heavy-tailed distributions and outliers, making them widely applicable in practice. We incorporate the dependence structure among predictors in the test statistics to enhance their powers. The limiting null distributions of the test statistics are derived to be the extreme value distribution of type I under regularity conditions. To reduce the size distortion, we further propose a multiplier bootstrap method based on the high-dimensional Gaussian approximations, which does not impose any structural assumptions on the unknown covariance matrices. We also evaluate the powers of proposed tests theoretically in comparison with two existing methods. The effectiveness of our proposed tests in the finite samples is illustrated through simulation studies and a real data application.