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Several experiments in high-energy physics and astrophysics can be treated as on/off measurements, where an observation potentially containing a new source or effect (on measurement) is contrasted with a background-only observation free of the effect (off measurement). In counting experiments, the significance of the new source or effect can be estimated with a widely-used formula from [LiMa], which assumes that both measurements are Poisson random variables. In this paper we study three other cases: i) the ideal case where the background measurement has no uncertainty, which can be used to study the maximum sensitivity that an instrument can achieve, ii) the case where the background estimate $b$ in the off measurement has an additional systematic uncertainty, and iii) the case where $b$ is a Gaussian random variable instead of a Poisson random variable. The latter case applies when $b$ comes from a model fitted on archival or ancillary data, or from the interpolation of a function fitted on data surrounding the candidate new source/effect. Practitioners typically use in this case a formula which is only valid when $b$ is large and when its uncertainty is very small, while we derive a general formula that can be applied in all regimes. We also develop simple methods that can be used to assess how much an estimate of significance is sensitive to systematic uncertainties on the efficiency or on the background. Examples of applications include the detection of short Gamma-Ray Bursts and of new X-ray or $gamma$-ray sources.
In this paper, after a discussion of general properties of statistical tests, we present the construction of the most powerful hypothesis test for determining the existence of a new phenomenon in counting-type experiments where the observed Poisson p
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Usually, equal time is given to measuring the background and the sample, or even a longer background measurement is taken as it has so few counts. While this seems the right thing to do, the relative error after background subtraction improves when m
A method to include multiplicative systematic uncertainties into branching ratio limits was proposed by M. Convery. That solution used approximations which are not necessarily valid. This note provides a solution without approximations and compares the results.