源强上限的计算
On Computing Upper Limits to Source Intensities
摘要：
天体物理学中的一个常见问题是确定一个源有多亮，而仍然没有被检测到。尽管这个问题可以简单地说出来，但解决方案涉及复杂的统计问题，需要仔细分析。与置信区间相反，这个概念从未被正式分析过，导致了各种各样的解决方案。在这里，我们制定和描述的问题，在一个自洽的方式。检测显著性通常由可接受的假阳性比例（I型错误）定义，我们调用假阴性的补充概念（II型错误），基于测试的统计能力，计算可检测源强度的上限。为了确定源必须具有的最小强度以使其被检测到，我们首先定义检测阈值，然后计算在给定阈值处检测到各种强度的源的概率。对应于指定的II类错误概率的强度定义了最小强度，并被标识为上限。因此，上限是探测程序的一个特征，而不是任何特定源的强度，不应与置信区间或源强度的其他估计混淆。这一点特别重要，因为越来越敏感的调查正在产生大量的目录。我们讨论了这些上限和置信区间之间的差异。这两种测量都是有用的量，应该报告，以便从目录中提取最多的科学，尽管它们回答了不同的统计问题：上限描述了源强度的推断范围，而上限校准了检测过程。我们提供了一个适用于所有检测算法的计算上限的方法。
英文摘要：
A common problem in astrophysics is determining how bright a source could be and still not be detected. Despite the simplicity with which the problem can be stated, the solution involves complex statistical issues that require careful analysis. In contrast to the confidence bound, this concept has never been formally analyzed, leading to a great variety of often ad hoc solutions. Here we formulate and describe the problem in a self-consistent manner. Detection significance is usually defined by the acceptable proportion of false positives (the TypeI error), and we invoke the complementary concept of false negatives (the TypeII error), based on the statistical power of a test, to compute an upper limit to the detectable source intensity. To determine the minimum intensity that a source must have for it to be detected, we first define a detection threshold, and then compute the probabilities of detecting sources of various intensities at the given threshold. The intensity that corresponds to the specified TypeII error probability defines that minimum intensity, and is identified as the upper limit. Thus, an upper limit is a characteristic of the detection procedure rather than the strength of any particular source and should not be confused with confidence intervals or other estimates of source intensity. This is particularly important given the large number of catalogs that are being generated from increasingly sensitive surveys. We discuss the differences between these upper limits and confidence bounds. Both measures are useful quantities that should be reported in order to extract the most science from catalogs, though they answer different statistical questions: an upper bound describes an inference range on the source intensity, while an upper limit calibrates the detection process. We provide a recipe for computing upper limits that applies to all detection algorithms.
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