The Significance of an Excess in a Counting Experiment: Assessing the Impact of Systematic Uncertainties and the Case with a Gaussian Background

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 Li & Ma, 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 a formula that 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 γ-ray sources. All the techniques presented in this paper are made available in a Python code that is ready to use.