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Confidence intervals for a binomial parameter or for the ratio of Poissonmeans are commonly desired in high energy physics (HEP) applications such asmeasuring a detection efficiency or branching ratio. Due to the discreteness ofthe data, in both of these problems the frequentist coverage probabilityunfortunately depends on the unknown parameter. Trade-offs among desideratahave led to numerous sets of intervals in the statistics literature, while inHEP one typically encounters only the classic intervals of Clopper-Pearson(central intervals with no undercoverage but substantial over-coverage) or afew approximate methods which perform rather poorly. If strict coverage isrelaxed, some sort of averaging is needed to compare intervals. In most of thestatistics literature, this averaging is over different values of the unknownparameter, which is conceptually problematic from the frequentist point of viewin which the unknown parameter is typically fixed. In contrast, we perform an(unconditional) {\it average over observed data} in the ratio-of-Poisson-meansproblem. If strict conditional coverage is desired, we recommendClopper-Pearson intervals and intervals from inverting the likelihood ratiotest (for central and non-central intervals, respectively). Lancaster's mid-$P$modification to either provides excellent unconditional average coverage in theratio-of-Poisson-means problem.

nuclear instruments and methods in physics research. section a, accelerators, spectrometers, detectors and associated equipment/nuclear instruments and methods in physics research. section a, accelerators, spectrometers, detectors and associated equipment

Frequentist Evaluation of Intervals Estimated for a Binomial Parameter and for the Ratio of Poisson Means