二项参数和Poisson均值比估计区间的频率估计
Frequentist Evaluation of Intervals Estimated for a Binomial Parameter and for the Ratio of Poisson Means
摘要：
在高能物理（HEP）应用（例如测量检测效率或分支比）中，通常需要二项式参数或泊松平均值比率的置信区间。由于数据的离散性，在这两个问题中，不幸的是，频率覆盖概率取决于未知参数。之间的权衡desiderata导致了众多的区间在统计文献中，而在HEP通常只遇到经典的Clopper-Pearson区间（中央区间没有覆盖不足，但大量的覆盖过度）或一些近似的方法，表现相当差。如果放宽严格覆盖，则需要某种平均来比较间隔。在大多数统计学文献中，这种平均是在未知参数的不同值上进行的，从频率论的角度来看，这在概念上是有问题的，其中未知参数通常是固定的。相比之下，我们在泊松均值比问题中执行（无条件）{ \it对观测数据求平均值}。如果需要严格的条件覆盖，我们推荐Clopper-Pearson区间和反转似然比检验的区间（分别用于中心和非中心区间）。Lancaster的中期$P$修改，提供了极好的无条件平均覆盖率的泊松比均值问题。
英文摘要：
Confidence intervals for a binomial parameter or for the ratio of Poisson means are commonly desired in high energy physics (HEP) applications such as measuring a detection efficiency or branching ratio. Due to the discreteness of the data, in both of these problems the frequentist coverage probability unfortunately depends on the unknown parameter. Trade-offs among desiderata have led to numerous sets of intervals in the statistics literature, while in HEP one typically encounters only the classic intervals of Clopper-Pearson (central intervals with no undercoverage but substantial over-coverage) or a few approximate methods which perform rather poorly. If strict coverage is relaxed, some sort of averaging is needed to compare intervals. In most of the statistics literature, this averaging is over different values of the unknown parameter, which is conceptually problematic from the frequentist point of view in which the unknown parameter is typically fixed. In contrast, we perform an (unconditional) {\it average over observed data} in the ratio-of-Poisson-means problem. If strict conditional coverage is desired, we recommend Clopper-Pearson intervals and intervals from inverting the likelihood ratio test (for central and non-central intervals, respectively). Lancaster's mid-$P$ modification to either provides excellent unconditional average coverage in the ratio-of-Poisson-means problem.
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