Asymptotic formulae for estimating statistical significance in particle physics analyses

Within the framework of likelihood-based statistical tests for particle physics measurements, we derive expressions for estimating the statistical significance of discovery using the asymptotic approximations of Wilks and Wald for four 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 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 significances using toy-based data in select cases and show the asymptotic formulae to be more computationally efficient than the toy-based approach. Additionally, different parameters within each measurement model are varied in order to understand their effect on the predicted significance.