Why the scaled and shifted t-distribution should not be used in the Monte Carlo method for estimating measurement uncertainty?

Abstract

The Supplement 1 (S1) to the GUM (Guide to the Expression of Uncertainty in Measurement) describes a general numerical approach, known as Monte Carlo method (MCM) for estimating measurement uncertainty, based on the principle of propagation of distributions. The MCM applies to a measurement model that has a single output quantity where the input quantities are characterized by specified PDFs (probability density function). When an input quantity has Type A uncertainty that is estimated with a limited number of observations (a sample), the GUM-S1 recommends using the scaled and shifted t-distribution conditioned on the sample mean and sample standard deviation as the PDF of the input quantity. This paper reveals that the scaled and shifted t-distribution is inappropriate for MCM because of the so-called t transformation distortion. This paper proposes an alternative PDF: a normal distribution conditioned on the sample mean and sample standard deviation. A calculation example is presented to demonstrate the inappropriateness of the scaled and shifted t-distribution and appropriateness of the proposed PDF for MCM. Two real-world examples are presented to compare the MCM based on the scaled and shifted t-distribution and the MCM based on the proposed PDF with several existing analytical methods.