Abstract
We give a classical confidence belt construction which unifies the treatment of upper confidence limits for null results and two-sided confidence intervals for non-null results. The unified treatment solves a problem (apparently not previously recognized) that the choice of upper limit or two-sided intervals leads to intervals which are not confidence intervals if the choice is based on the data. We apply the construction to two related problems which have recently been a battle-ground between classical and Bayesian statistics: Poisson processes with background, and Gaussian errors with a bounded physical region. In contrast with the usual classical construction for upper limits, our construction avoids unphysical confidence intervals. In contrast with some popular Bayesian intervals, our intervals eliminate conservatism (frequentist coverage greater than the stated confidence) in the Gaussian case and reduce it to a level dictated by discreteness in the Poisson case. We generalize the method in order to apply it to analysis of experiments searching for neutrino oscillations. We show that this technique both gives correct coverage and is powerful, while other classical techniques that have been used by neutrino oscillation search experiments fail one or both of these criteria.
Highlights
Classical confidence intervals are the traditional way in which high energy physicists report errors on results of experiments
There has been considerable dissatisfaction with the usual results of Neyman’s construction for upper confidence limits, in particular when the result is an unphysical interval. This dissatisfaction led the Particle Data Group (PDG) [2] to describe procedures for Bayesian interval construction in the troublesome cases: Poisson processes with background, and Gaussian errors with a bounded physical region
III, we review the troublesome cases of Poisson processes with background and Gaussian errors with a bounded physical region
Summary
Classical confidence intervals are the traditional way in which high energy physicists report errors on results of experiments. Approximate methods of confidence interval construction, in particular the likelihood-ratio method, are often used in order to reduce computation When these approximations are invalid, true confidence intervals can be obtained using the original (defining) construction of Neyman [1]. There has been considerable dissatisfaction with the usual results of Neyman’s construction for upper confidence limits, in particular when the result is an unphysical (or empty set) interval. This dissatisfaction led the Particle Data Group (PDG) [2] to describe procedures for Bayesian interval construction in the troublesome cases: Poisson processes with background, and Gaussian errors with a bounded physical region. The confidence level (C.L.) is more generally called α
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