Test for high-dimensional regression coefficients using refitted cross-validation variance estimation

Abstract

Testing a hypothesis for high-dimensional regression coefficients is of fundamental importance in the statistical theory and applications. In this paper, we develop a new test for the overall significance of coefficients in high-dimensional linear regression models based on an estimated U-statistics of order two. With the aid of the martingale central limit theorem, we prove that the asymptotic distributions of the proposed test are normal under two different distribution assumptions. Refitted cross-validation (RCV) variance estimation is utilized to avoid the overestimation of the variance and enhance the empirical power. We examine the finite-sample performances of the proposed test via Monte Carlo simulations, which show that the new test based on the RCV estimator achieves higher powers, especially for the sparse cases. We also demonstrate an application by an empirical analysis of a microarray data set on Yorkshire gilts.