Title: Power in High-Dimensional Testing Problems,
Abstract: Fan et al. (2015) recently introduced a remarkable 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 with the power enhancement principle.
However, when the dimensionality of the parameter space increases sufficiently
slowly with sample size and a marginal local asymptotic normality (LAN)
condition is satisfied, every test with asymptotic size smaller than one can be
improved with the power enhancement principle. While the marginal LAN condition
alone does not allow one to extend the latter statement to all rates at which
the dimensionality increases with sample size, we give sufficient conditions
under which this is the case.