A two-stage Monte Carlo approach to the expression of uncertainty with non-linear measurement equation and small sample size

Abstract

In this paper we propose a two-stage Monte Carlo procedure that explicitly takes into account the finite sample sizes used to estimate parameters of the input distributions of non-linear measurement equations. We will illustrate the approach with a case study involving the efficiency of a thermistor mount power sensor. The performance of the proposed approach will be compared with the standard GUM (US Guide to the Expression of Uncertainty in Measurement, American National Institute of Standards, ANSI/NCSL Z540-2-1997) approach for finite samples using simple linear and non-linear measurement equations. We will investigate performance in terms of coverage probabilities of derived coverage intervals for the mean of the measurand. Proposed Supplement 1 to the GUM (Guide to the Expression of Uncertainty in Measurement, Supplement 1, currently under review to the member organizations of the JCGM and the national measurement institutes (draft 3rd edn)) outlines a 'propagation of distributions' approach to deriving the distribution of a measurand for any non-linear function and for any set of random inputs. The Supplement's approach assumes that the distributions of the random inputs are known exactly. In practice, however, the distributions of the inputs will rarely be known exactly but must be estimated using possibly small samples. The two-stage Monte Carlo approach will be compared with the GUM Supplement's propagation of distribution approach for non-linear measurement equations. We will show that the GUM Supplement's approach underestimates uncertainty in the measurand if distributions estimated with small sample sizes are assumed to be known exactly.