BAYESIAN VARIANCE SEPARATION UNDER HETEROSCEDASTICITY - APPLICATION TO AN UNSTABLE MEASURAND

Let us analyse a measurement problem, where an unstable measurand is measured at n(i) different times - each time by (the same set of) n(j) devices or laboratories. The aim is to characterise the variabilities of the devices uncoupled from the variability of the measurand. We will formulate this measurement problem as a heteroscedastic linear mixed model (or alternatively as an n(j)-variate normal distribution) and develop a Bayesian approach using the non-informative Jeffreys prior to learn about the full distribution of the variance parameters. The method developed is applied to vacuum-measurement devices calibrated at the Physikalisch-Technische Bundesanstalt (PTB), in particular to pressure that was measured simultaneously by n(j) = 4 or n(j) = 6 devices and repeatedly n(i) = 16 times. The estimated variabilities for devices are robust and substantially smaller than the observed variances of their measurements. This case study demonstrates the full potential of variance separation under heteroscedasticity, e.g. for repeated simultaneous measurement of an unstable measurand.