Rank-based score tests for high-dimensional regression coefficients

Abstract

This article is concerned with simultaneous tests on linear regression coefficients in high-dimensional settings. When the dimensionality is larger than the sample size, the classic $F$-test is not applicable since the sample covariance matrix is not invertible. Recently, [5] and [17] proposed testing procedures by excluding the inverse term in $F$-statistics. However, the efficiency of such $F$-statistic-based methods is adversely affected by outlying observations and heavy tailed distributions. To overcome this issue, we propose a robust score test based on rank regression. The asymptotic distributions of the proposed test statistic under the high-dimensional null and alternative hypotheses are established. Its asymptotic relative efficiency with respect to [17]’s test is closely related to that of the Wilcoxon test in comparison with the $t$-test. Simulation studies are conducted to compare the proposed procedure with other existing testing procedures and show that our procedure is generally more robust in both sizes and powers.