Fixed Effects Bayesian Testing in High-Dimensional Linear Mixed Model

Hypothesis test on high-dimensional fixed effects is indispensable for investigating the utility of the predictors on response. In this case, the conventional frequentist methods for the case with fixed dimension fail completely due to the larger dimension than the sample size. Based on Bayes factor, a novel statistic is proposed for testing high-dimensional fixed effects. By transferring the linear mixed model, we build a bridge between Bayes factor and the mimic likelihood ratio statistic of the modified model, and provide a convictive justification for testing high-dimensional fixed effects through the mimic likelihood ratio statistic. The proposed statistic can be represented as the ratio of two quadratic forms constructed based on the random effects and random noises. In contrast to the existing results for quadratic forms based on i.i.d random variable sequence, we investigate the asymptotic normality for the quadratic form constructed by independently but not identically distributed random sequence. This theoretical result itself is very rewarding. To put the test procedure into practice, one-step iteration method is innovatively developed to determine the critical value. The power function under local alternatives is derived with some mild conditions. In numerical experiments, we demonstrate the higher powers in comparison with the existing method and the practical utility of the proposed method.