Statistical Models with Uncertain Error Parameters - "Errors On Errors" - and their Application to Particle Physics

In particle physics, data are collected and compared to theoretical predictions by constructing a statistical model that captures both the intrinsic stochastic nature of the measurement process and potential biases introduced by approximations in theoretical predictions and in the modeling of the experimental setup, referred to as systematic uncertainties. However, these uncertainties are often derived from limited data or estimated through ad-hoc procedures, introducing uncertainty in the uncertainty itself - informally referred to as ``error-on-error''. The Gamma Variance Model (GVM) is a statistical framework developed to address this issue by explicitly treating systematic uncertainties as imperfect estimates of unknown error parameters. This effectively incorporates errors-on-errors into the statistical model. A key practical consequence is that the GVM naturally reduces sensitivity to outliers and captures internal tensions within datasets by treating them as an additional source of uncertainty. In this thesis, the mathematical properties of the GVM are studied, and the model is extended to simplify its application in practical contexts, such as the combination of measurements and the fitting of physical parameters. These developments are applied to relevant physics examples, demonstrating the practical use of the model. Applications include a study of the 7-8 TeV ATLAS–CMS top-quark mass combination, an investigation of the W-boson mass discrepancy between the CDF result and the ATLAS and CMS measurements, and the fitting of parton distribution functions.