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.
翻译:许多常用的测试统计数据基于对无效假设证据的衡量标准。为了了解选择一项规范如何影响高维测试的功率特性,我们研究了在无限制参数空间的序列模型原型框架中,基于美元-诺姆的测试的一致性值,无限制参数空间,无效假设是所有观测都是零平均值的。此处将测试的一致性数据集定义为所有选择阵列的数据集,测试与参数空间差异的维度一致。我们定性了基于美元-诺尔姆测试的一致性,并发现,特别是,与一系列替代品的一致性不能仅以该替代品的美元-诺尔姆值来确定。我们的特征还揭示出出出一个出乎意料的单一性结果:即一致性数据集严格地以美元/年(0,\infty)美元增加,因此,基于较高美元测试的测试严格控制以较低美元/年一致性差值为基础的测试。这种单一性使我们能够建立新测试,在一致性行为方面,所有以美元-诺尔姆为主的测试都以不牺牲规模为基础。