Generalized asymptotic formulae for estimating statistical significance in high energy physics analyses

Within the framework of likelihood-based statistical tests for high energy physics measurements, we derive generalized expressions for estimating the statistical significance of discovery using the asymptotic approximations of Wilks and Wald for a variety of measurement models. These models include arbitrary numbers of signal regions, control regions, and Gaussian constraints. We extend our expressions to use the representative or "Asimov" dataset proposed by Cowan et al. such that they are made data-free. While many of the generalized expressions are complicated and often involve solving systems of coupled, multivariate equations, we show these expressions reduce to closed-form results under simplifying assumptions. We also validate the predicted significance using toy-based data in select cases.