We construct uncertainty intervals for weak Poisson signals in the presence of background. We consider the case where a primary experiment yields a realization of the signal plus background, and a second experiment yields a realization of the background. The data acquisitions times for the background-only experiment,T_bg, and the primary experiment,T, are selected so that their ratio varies from 1 to 25. The expected number of background counts in the primary experiment varies from 0.2 to 2. We construct 90 and 95 percent confidence intervals based on a propagation-of-errors method as well as two implementations of a Neyman procedure where acceptance regions are constructed based on a likelihood-ratio criterion that automatically determines whether the resulting confidence interval is one-sided or two-sided. The first Neyman procedure (due to Feldman and Cousins) neglects uncertainty in the background. In the other Neyman procedure, we account for uncertainty in the background with a parametric bootstrap method. We also construct minimum length Bayesian credibility intervals. For each method, we test for the presence of a signal based on the value of the lower endpoint of the uncertainty interval. When T_bg/T is 5 or more and the expected background is 2 or less, the Feldman Cousins method outperforms the other methods considered.
Evaluation of three methods for calculating statistical significance when incorporating a systematic uncertainty into a test of the background-only hypothesis for a Poisson process
Comment on "Including Systematic Uncertainties in Confidence Interval Construction for Poisson Statistics"
Conrad等人最近在一篇论文（Physical Review D 67（2003）012002）中讨论了将系统不确定性纳入置信区间计算的问题。在他们的工作中，探测器效率和背景通量预测的系统不确定性被纳入以下的混合频率-贝叶斯公式的Feldman和Highland，但使用的似然比排序的Feldman和Feldman，以产生“统一”的置信区间。一般来说，所得到的间隔表现得与人们直观预期的一样，即随着不确定性的增加而增加。然而，据指出，对于观测到的事件数量少于预期背景或与预期背景数量相当的情况，间隔有时可能以完全违反直觉的方式表现-在不确定性增加的情况下，最初的间隔会减少，但这仅限于信号效率不确定性增加的情况。在这篇评论中，我们表明，有问题的行为是由于整合的信号效率的不确定性，同时最大限度地提高最佳拟合备择假设的可能性。如果备择假设似然是通过无条件最大化未知信号和信号效率不确定性来确定的，则限值显示正确的直观行为。
