A Novel Approach of High Dimensional Linear Hypothesis Testing Problem

This paper proposes an innovative double power-enhanced testing procedure for inference on high-dimensional linear hypotheses in high-dimensional regression models. Through a projection approach that aims to separate useful inferential information from the nuisance one, our proposed test accurately accounts for the impact of high-dimensional nuisance parameters. We discover that with a carefully-designed projection matrix, the projection procedure enables us to transform the problem of interest into a test on moment conditions, from which we construct a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi></mml:math> -statistic-based test that is applicable in simultaneous inference on a diverging number of linear hypotheses. We prove that under regularity conditions, the plug-in test statistic converges to its oracle counterpart, acting as well as if the nuisance parameters were known in advance. Moreover, we introduce an implementation-friendly version to tackle the computational challenge. Asymptotic null normality is established to provide convenient tools for statistical inference, accompanied by rigorous power analysis. To further strengthen the testing power, we develop two power enhancement techniques to boost the power from two distinct aspects respectively, and integrate them into one powerful testing procedure to achieve double power enhancement. The finite-sample performance is demonstrated using simulation studies, and an empirical analysis of a real data example.