Tests for high-dimensional generalized linear models under general covariance structure

This study investigates the testing of regression coefficients within high-dimensional generalized linear models featuring general covariance structures. The derived asymptotic properties reveal that distinct covariance structures can lead to varying limiting null distributions, including the normal distribution, for a widely employed quadratic-norm based test statistic. This circumstance renders it infeasible to determine critical values through a limiting null distribution. In response to this challenge, we propose a multiplier bootstrap test procedure for practical implementation. Additionally, we introduce a modified version of this procedure, incorporating projection when dealing with nuisance parameters. We then proceed to examine the asymptotic level and power of the proposed tests and assess their finite-sample performance through simulations. Finally, we present a real data analysis to illustrate the practical application of the proposed tests.