Confidence interval performance is typically assessed in terms of two criteria: coverage probability and interval width (or margin of error). In this work we detail the importance of defining the margin of error relative to the magnitude of the estimated proportion when the success probability is small. We compare the performance of four common proportion estimators: the Wald, Clopper-Pearson, Wilson and Agresti-Coull, in the context of rare-event probabilities. We show that incompatibilities between the margin of error and the proportion results in very narrow intervals (requiring extremely large sample sizes), or intervals that are too wide to be practically useful. We propose a relative margin of error scheme that is consistent with the order of magnitude of the proportion. Confidence interval performance is thus assessed in terms of satisfying this relative margin of error, in conjunction with achieving a desired coverage probability. We show that when adherence to this relative margin of error is considered as a requirement for satisfactory interval performance, all four interval estimators perform somewhat similarly for a given sample size and confidence level. The proposed relative margin of error scheme is evaluated by way of its application to a number of recent studies from the literature.
翻译:在这项工作中,我们详细说明了在成功概率小的情况下根据估计比例的幅度来界定误差幅度的重要性。我们比较了四个共同比例估计值的性能:Wald、Clopper-Pearson、Wilson和Agresti-Coull,在稀有事件概率方面;我们显示误差幅度与比例结果之间在非常狭窄的间隔(需要极大样本大小)或间隔太广而实际上无实际用处的间隔上的不相容性。我们提出了与比例大小相符的相对误差幅度办法。因此,在满足这一相对误差幅度的同时,在达到预期的概率方面评估了信任间隔性能。我们表明,如果将遵守这一相对误差幅度视为令人满意的间隔性能的一项要求,那么所有四个间隔估计值对于某一抽样规模和信任程度而言都具有某种相似性能。拟议的误差相对比率办法通过应用最近从文献中得出的一些研究加以评估。