Unified approach to the classical statistical analysis of small signals

TLDR

The authors introduce a unified classical confidence belt that simultaneously handles upper limits for null results and two‑sided intervals for non‑null results, and extend it to neutrino‑oscillation experiments. They construct the belt, apply it to Poisson processes with background and Gaussian errors with bounded physical regions, and generalize it for neutrino‑oscillation search analyses. The approach yields correct frequentist coverage, avoids unphysical or overly conservative intervals, and outperforms existing classical methods in neutrino‑oscillation searches.

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 battleground 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.