New Tests for High-Dimensional Linear Regression Based on Random Projection

Abstract

We consider the problem of detecting the significance in high-dimensional linear models, allowing the dimension of the regression coefficient to be greater than the sample size. We propose novel test statistics for the hypothesis testing of testing the global significance of the linear model as well as the significance of part of the regression coefficients. The new tests are based on randomly projecting high-dimensional data into a space of low dimensions and then working with the classical F-test using the projected data. An appealing feature of the proposed tests is that they have a simple form and are computationally easy to implement. We derive the asymptotic local power functions of the proposed tests and compare with the existing methods for hypothesis testing in high-dimensional linear models. We also provide a sufficient condition under which our proposed tests have higher asymptotic relative efficiency. Through simulation studies, we evaluate the finite-sample performances of the proposed tests and demonstrate that it performs better than the existing tests in the models we considered. Applications to real high-dimensional gene expression data are also provided for illustration.