信号显著性的改进推断
Improved Inference for the Signal Significance
摘要：
我们研究了几种基于似然的统计特性，这些特性通常用于在具有已知背景但未知信号分数的混合模型下测试已知信号的存在。在无信号的零假设下，所有统计量在大样本中遵循标准正态分布，但在低样本量下可能会出现实质性偏差。近似为各自的$p$值推导出各种订单的准确性使用的方法埃奇沃思展开。研究了对正态性的遵守，并根据所产生的通货膨胀或通货紧缩量化了偏差的大小。我们发现，近似到三阶精度一般足以保证$p$ -值与标称假阳性错误率在五西格玛范围内（$p$ -值$= 2.87 \times 10^{-7}$）的经典沃尔德，分数，和似然比（LR）统计在相对较低的样本。LR不仅具有更好的正态性，而且在假阴性错误率方面始终优于所有其他统计数据。其原因被证明是与高阶累积量的行为收集从四阶Edgeworth展开。最后，本文提出了一个保守的方法，用于有限样本的调整，同时用随机场理论（也称为“随机场”）来解释“望他处”效应。Gross-Vitells方法）。
英文摘要：
We study the properties of several likelihood-based statistics commonly used in testing for the presence of a known signal under a mixture model with known background, but unknown signal fraction. Under the null hypothesis of no signal, all statistics follow a standard normal distribution in large samples, but substantial deviations can occur at low sample sizes. Approximations for respective $p$-values are derived to various orders of accuracy using the methodology of Edgeworth expansions. Adherence to normality is studied, and the magnitude of deviations is quantified according to resulting inflation or deflation. We find that approximations to third-order accuracy are generally sufficient to guarantee $p$-values with nominal false positive error rates in the five sigma range ($p$-value $= 2.87 \times 10^{-7}$) for the classic Wald, score, and likelihood ratio (LR) statistics at relatively low samples. Not only does LR have better adherence to normality, but it also consistently outperforms all other statistics in terms of false negative error rates. The reasons for this are shown to be connected with high-order cumulant behavior gleaned from fourth order Edgeworth expansions. Finally, a conservative procedure is suggested for making finite sample adjustments while accounting for the look elsewhere effect with the theory of random fields (a.k.a. the Gross-Vitells method).
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