Simultaneous test for linear model via projection

In this paper, we focus on the simultaneous test for the high-dimensional linear model coefficient. In high-dimensional setting, the traditional F-test is infeasible while the dimension is larger than the sample size because the sample covariance matrix is not invertible. Even when the dimension is smaller than the sample size, the efficiency of the classical F-test diminishes rapidly as the dimensionality increases. Several existing methods modified the classical F-test by removing the inverse term in the statistics and performed well. However, those modifications may lose efficiency due to the ignorance of correlation and those moment based procedures are sensitive to outlying observations and heavy tailed data. We propose a centered rank based method using projection which is robust in a broad range of distribution and alternative hypothesis. Under some mild conditions, we derive the best projection direction. The proposed procedure can retain type I error rate pretty well with the asymptotic distribution due to the data splitting strategy. Finite sample simulation studies are in accordance with the theoretical results.