Power-Constrained Limits

We propose a method for setting limits that avoids excluding parameter values for which the sensitivity falls below a specified threshold. These “power-constrained” limits (PCL) address the issue that motivated the widely used CLs procedure, but do so in a way that makes more transparent the properties of the statistical test to which each value of the parameter is subjected. A case of particular interest is for upper limits on parameters that are proportional to the cross section of a process whose existence is not yet established. The basic idea of the power constraint can easily be applied, however, to other types of limits.

💡 Research Summary

The paper addresses a well‑known problem in high‑energy‑physics searches: the tendency of standard frequentist confidence‑interval constructions to exclude parameter values for which the experiment has essentially no sensitivity. This “spurious exclusion” arises when the signal strength μ is so small that the probability distributions of the test statistic under the signal hypothesis and under the background‑only hypothesis (μ = 0) are nearly indistinguishable. In such cases a 5 %‑level test can still reject the signal hypothesis simply because of a statistical fluctuation, leading to limits that are stronger than justified by the data.

The widely used CLs method was introduced to mitigate this problem. CLs defines a modified p‑value ratio CLs = pμ / (1 – p0) and excludes μ only if CLs < α. Because CLs is always larger than the ordinary p‑value pμ, the probability of exclusion under the signal hypothesis is reduced, making the limits conservative. However, this conservatism also means that the actual coverage of CLs limits is typically well above the nominal confidence level, and the coverage is not uniform across the parameter space.

The authors propose a different solution called Power‑Constrained Limits (PCL). The key idea is to use the statistical power of the test with respect to the background‑only alternative, denoted M0(μ) = P(reject μ | μ = 0), as a quantitative measure of sensitivity. A minimum power threshold Mmin is chosen (the authors recommend Mmin = Φ(‑1) ≈ 0.1587, i.e. the one‑sigma point of a standard normal). For any value of μ whose power falls below this threshold, the experiment is deemed “insensitive” and the corresponding μ is automatically retained in the confidence interval, regardless of the outcome of the test. For μ with power ≥ Mmin, the usual α‑level test is applied: μ is excluded if the observed p‑value pμ < α. The resulting interval is the union of all μ that are either (i) not rejected by the test or (ii) rejected but have insufficient power. Consequently, the coverage is 100 % for all μ with M0(μ) < Mmin and exactly (1 – α) for μ with M0(μ) ≥ Mmin.

The paper works out the PCL construction in detail for the common case of a Gaussian‑distributed estimator \hat{μ} with known standard deviation σ. The critical region for an upper‑limit test is \hat{μ} < μ – σ Φ⁻¹(1 – α). The power under the background‑only hypothesis is M0(μ) = Φ(μ/σ – Φ⁻¹(1 – α)). Solving M0(μ) ≥ Mmin yields a minimum detectable signal strength μmin = σ