Variance-estimation-free test of significant covariates in high-dimensional regression

Abstract

In a high-dimensional linear regression model, this article is concerned with testing statistical significance of a subset of regression coefficients. The conventional partial F-test is not applicable in high-dimensional situations. Several methods for testing whether any of the discarded covariates is significant conditional on relative importance of predictors have been proposed in the recent literature, but they are adversely affected by the overestimation of the variance. To overcome this issue, we propose a novel nonparametric testing procedure to avoid this problem and enhance the empirical power. In addition, the new test is very effective when error distribution deviates from the normal scenario and can integrate all the individual information of the discarded covariates. Under the high-dimensional null and alternative hypotheses, we derive the asymptotic distribution of the proposed test statistic, which allows power evaluation of the test. Numerical studies are carried out to examine the numerical performance of the test. The results show that the test proposed here behaves well in terms of sizes and power and significantly outperforms the existing choices in a range of settings.