A Bayesian-motivated test for high-dimensional linear regression models with fixed design matrix

This paper considers testing regression coefficients in high-dimensional linear model with fixed design matrix. This problem is highly irregular in the frequentist point of view. In fact, we prove that no test can guarantee nontrivial power even when the true model deviates greatly from the null hypothesis. Nevertheless, Bayesian methods can still produce tests with good average power behavior. We propose a new test statistic which is the limit of Bayes factors under normal distribution. The null distribution of the proposed test statistic is approximated by Lindeberg's replacement trick. Under certain conditions, the global asymptotic power function of the proposed test is also given. The finite sample performance of the proposed test is demonstrated via simulation studies.