  As a common step in refining their scientific inquiry, investigators are
often interested in performing some screening of a collection of given
statistical hypotheses. For example, they may wish to determine whether any one
of several patient characteristics are associated with a health outcome of
interest. Existing generic methods for testing a multivariate hypothesis --
such as multiplicity corrections applied to individual hypothesis tests -- can
easily be applied across a variety of problems but can suffer from low power in
some settings. Tailor-made procedures can attain higher power by building
around problem-specific information but typically cannot be easily adapted to
novel settings. In this work, we propose a general framework for testing a
multivariate point null hypothesis in which the test statistic is adaptively
selected to provide increased power. We present theoretical large-sample
guarantees for our test under both fixed and local alternatives. In simulation
studies, we show that tests created using our framework can perform as well as
tailor-made methods when the latter are available, and we illustrate how our
procedure can be used to create tests in two settings in which tailor-made
methods are not currently available.
