Power in High-Dimensional Testing Problems

Fan et al. (2015) recently introduced a method for increasing asymptotic power of tests in high-dimensional testing problems. If applicable to a given test, their power enhancement principle leads to an improved test that has the same asymptotic size, uniformly non-inferior asymptotic power, and is consistent against a strictly broader range of alternatives than the initially given test. We study under which conditions this method can be applied and show the following: In asymptotic regimes where the dimensionality of the parameter space is fixed as sample size increases, there often exist tests that can not be further improved by the power enhancement principle. When the dimensionality can increase with sample size, however, there typically is a range of slowly diverging rates for which every test with asymptotic size smaller than one can be improved with the power enhancement principle. We also address under which conditions the latter statement even extends to all rates at which the dimensionality increases with sample size.