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 <i>errors-on-errors</i>). 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>±</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> or 0), and we then generalise the nuisance-parameter approach to include <i>errors-on-errors</i>. We then present analytical formulas for estimating central values, confidence intervals, and goodness-of-fit when <i>errors-on-errors</i> 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.