Higher criticism (HC) is a popular method for large-scale inference problems based on identifying unusually high proportions of small pvalues. It has been shown to enjoy a lower-order optimality property in a simple normal location mixture model which is shared by the ‘tailor-made’ parametric generalised likelihood ratio test (GLRT) for the same model, however HC has also been shown to perform well outside this ‘narrow’ model. We develop a higher-order framework for analysing the power of these and similar procedures, which reveals the perhaps unsurprising fact that the GLRT enjoys an edge in power over HC for the normal location mixture model. We also identify a similar parametric mixture model to which HC is similarly ‘tailor-made’ and show that the situation is (at least partly) reversed there. We also show that in the normal location mixture model a procedure based on the empirical moment-generating function enjoys the same local power properties as the GLRT and may be recommended as an easy to implement (and interpret), complementary procedure to HC. Some other practical advice regarding the implementation of these procedures is provided. Finally we provide some simulation results to help interpret our theoretical findings.

中文翻译：

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超越 HC：对稀有/弱替代品进行更敏感的测试

高级批评 (HC) 是基于识别异常高比例的小 pvalue 的大规模推理问题的流行方法。它已被证明在一个简单的正常位置混合模型中具有低阶最优性，该模型由同一模型的“量身定制”参数广义似然比检验 (GLRT) 共享，但是 HC 也被证明可以执行远远超出这种“狭隘”模式。我们开发了一个高阶框架来分析这些和类似程序的能力，这揭示了一个可能不足为奇的事实，即 GLRT 在正常位置混合模型的能力上优于 HC。我们还确定了一个类似的参数混合模型，HC 与之类似地“量身定制”，并表明情况（至少部分）在那里发生了逆转。我们还表明，在正常位置混合模型中，基于经验矩生成函数的程序享有与 GLRT 相同的局部功率特性，并且可能被推荐为易于实施（和解释）的 HC 补充程序。还提供了有关实施这些程序的一些其他实用建议。最后，我们提供了一些模拟结果来帮助解释我们的理论发现。