  Many commonly used test statistics are based on a norm measuring the evidence
against the null hypothesis. To understand how the choice of a norm affects
power properties of tests in high dimensions, we study the consistency sets of
$p$-norm based tests in the prototypical framework of sequence models with
unrestricted parameter spaces, the null hypothesis being that all observations
have zero mean. The consistency set of a test is here defined as the set of all
arrays of alternatives the test is consistent against as the dimension of the
parameter space diverges. We characterize the consistency sets of $p$-norm
based tests and find, in particular, that the consistency against an array of
alternatives cannot be determined solely in terms of the $p$-norm of the
alternative. Our characterization also reveals an unexpected monotonicity
result: namely that the consistency set is strictly increasing in $p \in (0,
\infty)$, such that tests based on higher $p$ strictly dominate those based on
lower $p$ in terms of consistency. This monotonicity allows us to construct
novel tests that dominate, with respect to their consistency behavior, all
$p$-norm based tests without sacrificing size.
