Consistency of $p$-norm based tests in high-dimensions: characterization, monotonicity, domination

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 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 can not 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 asymptotic size.