Silent coverage failures in rare-event searches and a degeneracy index that predicts them

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 $\mathcal I_{\mathrm{PF}}(\delta\nu;\vartheta_0)=(\beta,\gamma)$, which maps a specified expected-count deformation, after projection onto the complete fitted tangent space, to $\beta$, the signed fitted signal shift in profiled standard-error units, and $\gamma$, the Poisson--Fisher norm of the unabsorbed residual. Locally, the sign of $\beta$ identifies the threatened endpoint, while larger $\gamma$ 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 $|\beta|$ and small $\gamma$ 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.