Tests based on the $2$- and $\infty$-norm have received considerable attention in high-dimensional testing problems, as they are powerful against dense and sparse alternatives, respectively. The power enhancement principle of Fan et al. (2015) combines these two norms to construct tests that are powerful against both types of alternatives. Nevertheless, the $2$- and $\infty$-norm are just two out of the whole spectrum of $p$-norms that one can base a test on. In the context of testing whether a candidate parameter satisfies a large number of moment equalities, we construct a test that harnesses the strength of all $p$-norms with $p\in[2, \infty]$. As a result, this test consistent against strictly more alternatives than any test based on a single $p$-norm. In particular, our test is consistent against more alternatives than tests based on the $2$- and $\infty$-norm, which is what most implementations of the power enhancement principle target. We illustrate the scope of our general results by using them to construct a test that simultaneously dominates the Anderson-Rubin test (based on $p=2$) and tests based on the $\infty$-norm in terms of consistency in the linear instrumental variable model with many (weak) instruments.

翻译：基于 $2$-范数与 $\infty$-范数的检验在高维检验问题中受到了广泛关注，因为它们分别对稠密备择与稀疏备择具有较高的功效。Fan 等人（2015）提出的功效增强原理结合了这两种范数，以构建对两类备择均具有功效的检验。然而，$2$-范数与 $\infty$-范数仅是可用于构建检验的整个 $p$-范数谱系中的两种。在检验候选参数是否满足大量矩等式的背景下，我们构建了一种检验方法，它利用了所有 $p \in [2, \infty]$ 的 $p$-范数的优势。因此，该检验比任何基于单一 $p$-范数的检验能一致地对抗更广泛的备择假设。特别地，我们的检验比基于 $2$-范数与 $\infty$-范数的检验能一致地对抗更多备择，而后者正是大多数功效增强原理实现方案所针对的目标。我们通过在线性工具变量模型中使用大量（弱）工具变量的情形，构建了一个在一致性方面同时优于 Anderson-Rubin 检验（基于 $p=2$）和基于 $\infty$-范数检验的示例，以此阐释了我们一般性结果的适用范围。