基于$p$ -范数的高维检验的一致性：特征化，单调性，支配性
Consistency of $p$-norm based tests in high dimensions: characterization, monotonicity, domination
摘要：
许多常用的检验统计量都是基于一个标准来衡量反对原假设的证据。要了解如何选择一个规范的影响力的测试在高维属性，我们研究了一致性集的$p$ -规范为基础的测试在原型框架的序列模型与不受限制的参数空间，原假设是所有的观察都是零均值。在这里，测试的一致性集被定义为当参数空间的维度发散时，测试与之一致的所有备选方案阵列的集合。我们的特点的一致性集的$p$ -规范为基础的测试，并发现，特别是，对一系列的替代品的一致性不能单独确定的$p$ -规范的替代品。我们的表征还揭示了一个意想不到的单调性结果：即一致性集是严格增加的$p \in (0, \infty)$，这样的测试的基础上更高的$p$严格支配那些基于较低的$p$的一致性。这种单调性使我们能够构建新的测试，占主导地位，就其一致性的行为，所有$p$规范为基础的测试，而不牺牲大小。
英文摘要：
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.
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