Testing high-dimensional regression coefficients in linear models via Bayes factor

This paper considers the problem of testing for the overall significance of a large number of remaining predictors in high-dimensional linear models, given that a few predictors are known to have significant effect on the response. We propose a novel test based on Bayes factor, assuming that the errors are normally distributed. The proposed test is also applicable to testing the global significance of the linear models. The asymptotic normality of the Bayes factor-based test statistic under the null hypothesis and the local alternatives is established. Furthermore, we derive the asymptotic local power function of the proposed test and compare it with that of existing methods. Monte Carlo simulation results show that the limiting approximation to the null distribution is highly accurate in finite samples, and the proposed test outperforms several well-known tests in the literature in terms of Type I error rate and the empirical power. The practical effectiveness of the proposed test is illustrated through two real-data applications.