Analysis of approximations of GUM supplement 2 based non-Gaussian PDFs of measurement models with Rosenblatt Gaussian transformation mappings

Abstract

In scientific metrology practise the application of Monte Carlo simulations with the aid of the GUM Supplement 2 (GS2) technique for performing multivariate uncertainty analyses is now more prevalent, however a key remaining challenge for metrologists in many laboratories is the implicit assumption of Gaussian characteristics for summarizing and analysing measurement model results. Whilst non-Gaussian probability density functions (PDFs) may result from Monte Carlo simulations when the GS2 is applied for more complex non-linear measurement models, in practice results are typically only reported in terms of multivariate expected and covariance values. Due to this limitation the measurement model PDF summary is implicitly restricted to a multivariate Gaussian PDF in the absence of additional higher order statistics (HOS) information. In this paper an earlier classical theoretical result by Rosenblatt that allows for an arbitrary multivariate joint distribution function to be transformed into an equivalent system of Gaussian distributions with mapped variables is revisited. Numerical simulations are performed in order to analyse and compare the accuracy of the equivalent Gaussian system of mapped random variables for approximating a measurement model’s PDF with that of an exact non-Gaussian PDF that is obtained with a GS2 Monte Carlo statistical simulation. Results obtained from the investigation indicate that a Rosenblatt transformation offers a convenient mechanism to utilize just the joint PDF obtained from the GS2 data in order to both sample points from a non-Gaussian distribution, and also in addition which allows for a simple two-dimensional approach to estimate coupled uncertainties of random variables residing in higher dimensions using conditional densities without the need for determining parametric based copulas.