Correlated Systematic Uncertainties and Errors-on-Errors in Measurement Combinations: Methodology and Application to the 7-8 TeV ATLAS-CMS Top Quark Mass Combination

The Gamma Variance Model (GVM) 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 derive useful relations that allow one to connect the BLUE and nuisance-parameter methods when the correlations induced by systematic uncertainties are non-trivial (1, -1 or 0), and we generalise the nuisance-parameter approach to include errors-on-errors. We then illustrate some of the properties of the GVM by applying it to the 7-8 TeV ATLAS-CMS top quark mass combination. We present results by considering the largest systematic uncertainties as uncertain, one at a time, and we vary their associated error-on-error parameters. This procedure is useful for identifying the systematic uncertainties to which a combination is sensitive when they are themselves uncertain. We also explore the hypothetical scenario of including an outlier in the combination, which could become relevant for future combinations, by artificially adding a fictitious measurement to it. This example highlights a key feature of the GVM: its sensitivity to the internal consistency of the input data.