The likelihood ratio test (LRT) and the related $F$ test, do not (even asymptotically) adhere to their nominal $\chi^2$ and $F$ distributions in many statistical tests common in astrophysics, thereby casting many marginal line or source detections and non-detections into doubt. Although there are many legitimate uses of these statistics, in some important cases it can be impossible to compute the correct false positive rate. For example, it has become common practice to use the LRT or the $F$ test for detecting a line in a spectral model or a source above background despite the lack of certain required regularity conditions. In these and other settings that involve testing a hypothesis that is on the boundary of the parameter space, {\it contrary to common practice, the nominal $\chi^2$ distribution for the LRT or the $F$ distribution for the $F$ test should not be used}. In this paper, we characterize an important class of problems where the LRT and the $F$ test fail and illustrate this non-standard behavior. We briefly sketch several possible acceptable alternatives, focusing on Bayesian posterior predictive probability-values. We present this method in some detail, as it is a simple, robust, and intuitive approach. This alternative method is illustrated using the gamma-ray burst of May 8, 1997 (GRB 970508) to investigate the presence of an Fe K emission line during the initial phase of the observation.
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机译:在天体物理学中许多常见的统计检验中,似然比检验(LRT)和相关的$ F $检验都(甚至渐近地)不遵循其名义上的\\ chi ^ 2 $和$ F $分布,从而投射了许多边际线或源检测和未检测产生怀疑。尽管这些统计信息有许多合法用途,但在某些重要情况下,可能无法计算出正确的误报率。例如,尽管缺少某些必需的规律性条件,但使用LRT或$ F $测试来检测光谱模型中的一条线或本底以上的信号源已成为一种常见的做法。在涉及测试参数空间边界上的假设的这些设置和其他设置中,{\与常规做法相反,LRT的名义$ \ chi ^ 2 $分布或$ F的$ F $分布$ test不应该使用}。在本文中,我们描述了LRT和$ F $测试失败的一类重要问题,并说明了这种非标准行为。我们简要地概述了几种可能的可接受的替代方法,重点放在贝叶斯后验预测概率值上。我们将详细介绍此方法,因为它是一种简单,健壮和直观的方法。使用1997年5月8日的伽马射线暴(GRB 970508)说明了该替代方法,以研究观测初始阶段Fe K发射谱线的存在。
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