🤖 AI Summary
In searches for rare events, model misspecification can severely compromise the coverage of confidence intervals, particularly when the signal is non-negative and event counts are extremely low, rendering such issues difficult to detect. This work proposes the Poisson–Fisher degradation index $\mathcal{I}_{\mathrm{PF}}(\delta\nu; \vartheta_0)=(\beta,\gamma)$, integrating profile likelihood, Poisson–Fisher geometry, and tangent space projection to quantify how model bias affects the coverage of upper limits and discovery sensitivity. The analysis reveals that positive bias yields conservative upper limits but degrades discovery-side coverage, whereas negative bias may lead to undercoverage of upper limits. Deformations with low detectability yet high bias can evade standard diagnostics and require auxiliary constraints for identification. Imposing collinearity constraints to model such deformations restores nominal coverage across grid points at the cost of reduced sensitivity.
📝 Abstract
Searches for new physics in low-background experiments infer a non-negative signal strength from few events and often report an upper limit. Nominal frequentist coverage requires both a valid interval construction and an adequate data model. We study how controlled model departures affect lower- and upper-endpoint coverage for six interval procedures in an exact Poisson counting experiment, dark-matter recoil spectra, and a neutrinoless-double-beta-decay peak search. We introduce the Poisson--Fisher degeneracy index $\mathcal I_{\mathrm{PF}}(δν;\vartheta_0)=(β,γ)$, which maps a specified expected-count deformation, after projection onto the complete fitted tangent space, to $β$, the signed fitted signal shift in profiled standard-error units, and $γ$, the Poisson--Fisher norm of the unabsorbed residual. Locally, the sign of $β$ identifies the threatened endpoint, while larger $γ$ implies greater detectability by the saturated-Poisson goodness-of-fit test used here at a fixed $5\%$ type-I error rate. Across the studied deformations, positive signal-like bias degrades discovery-side coverage while making upper limits conservative; negative signal bias from overestimated signal efficiency can make the upper endpoint undercover. Calibration under the nominal simulator does not protect against misspecification of that simulator relative to the data-generating process. A plausible deformation with large $|β|$ and small $γ$ may therefore evade diagnosis and should be represented by a nuisance constrained with auxiliary information or included in a defensible envelope. In the exactly collinear constrained-nuisance benchmark, modelling the deformation restores coverage at the evaluated grid points, at a quantifiable cost in interval sensitivity.