Correlated systematic uncertainties and errors-on-errors in measurement combinations with an application to the 7–8 TeV ATLAS–CMS top quark mass combination

The Gamma Variance Model is a statistical model that incorporates uncertainties in the assignment of systematic errors (informally called errors-on-errors). The model is of particular use in analyses that combine the results of several measurements. In the past, combinations have been carried out using two alternative approaches: the Best Linear Unbiased Estimator (BLUE) method or what we will call the nuisance-parameter method. In this paper, we obtain a general relation between the BLUE and nuisance-parameter methods when the correlations induced by systematic uncertainties are non-trivial (i.e., not ±1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm 1$$\end{document} or 0), and we then generalise the nuisance-parameter approach to include errors-on-errors. We then present analytical formulas for estimating central values, confidence intervals, and goodness-of-fit when errors-on-errors are incorporated into the statistical model. To illustrate the properties of the Gamma Variance Model, we apply it to the 7–8 TeV ATLAS–CMS top quark mass combination. We also explore a hypothetical scenario by artificially adding a fictitious measurement as an outlier to the combination, illustrating a key feature of the Gamma Variance Model – its sensitivity to the internal consistency of the input data – which could become relevant for future combinations.