Abstract
Testing a global hypothesis for a set of variables is a fundamental problem in statistics with a wide range of applications. A few well-known classical tests include the Hotelling's T2 test, the F test, and the empirical Bayes based score test. These classical tests, however, are not robust to the signal strength and could have a substantial loss of power when signals are weak or moderate, a situation we commonly encounter in contemporary applications. In this paper, we propose a Minimax Optimal Ridge-type Set Test (MORST), a simple and generic method for testing a global hypothesis. The power of MORST is robust and considerably higher than that of the classical tests when the strength of signals is weak or moderate. In the meantime, MORST only requires a slight increase in computation compared to these existing tests, making it applicable to the analysis of massive genome-wide data. We also provide the generalizations of MORST that are parallel to the traditional Wald test and Rao's score test in asymptotic settings. Extensive simulations demonstrated the robust power of MORST and that the type I error of MORST was well controlled. We applied MORST to the analysis of the whole-genome sequencing data from the Atherosclerosis Risk in Communities (ARIC) study, where MORST detected 20%-250% more signal regions than the classical tests.