## 4 General Methodology for Monte Carlo Simulation  
In the classical GUM method, two types of calculations are specified: the estimated value of the measurand and the typical uncertainty estimation, both calculations are separated and independent [1]. In GUM-S1, the simulation with random samples does not allow such separation [8]. According to [8], GUM-S1 provides a general approach to obtain an approximate numerical representation \(G\) of the distribution function \(G_{Y}\) for the measurand \(Y\). The heart of the approach is repeated sampling from the _PDFs_ for the input quantities \(Xi\) and the evaluation of the model in each case. The general description of the MCM approach would be as follows:  
1. Select the number \(M\) of Monte Carlo trails to be made. 2. Generate \(M\) vectors, by sampling from the assigned _PDFs_, as realizations of the input quantities (\(X_{i}\)). 3. For each such vector, form the corresponding model value of \(Y\), yielding \(M\) model values. 4. Sort these \(M\) models in strictly increasing order to provide \(G\). 5. Use \(G\) to form an estimate \(y\) of \(Y\) and the standard uncertainty \(u(y)\) associated with \(y\). 6. Use \(G\) to form an appropriate coverage interval for Y, for a stipulated coverage probability p, computing \(y_{low}\) and \(y_{high}\) being \(y_{low}\) is 2.5 th percentile (P2.5) and \(y_{high}\) 97.5 th percentile (P97.5), if a 95% confidence level is desired. When the _PDF_ is not symmetrical, the uncertainty is not symmetrical either and the coverage factor is no longer meaningful. In fact, the coverage factor \(k\) is not considered in GUM-S1. In GUM-S1, the expanded uncertainty is directly calculated with the approximate numerical representation of the _PDF_ (\(G_{Y}\)) and there is no need to assume certain hypotheses on the normality or the effective degrees of freedom in a Student's t-distribution. If the MCM approach is used only for estimating the uncertainty, it must be proven that the measurand value \(y\) given by the MCM is compatible with the result given by the classical GUM method. When applying MCM the sample size \(M\) must be chosen. If the sample size is not large enough, different results could be obtained when repeating the simulation. According to GUM-S1, \(M\) could be in the order of \(10^{6}\) but there is no guarantee that this or any specific pre-assigned number will suffice [8]. In this work, the adaptive Monte Carlo method (AMCM) was selected as this procedure controls the repeatability of the result and optimizes \(M\), though its programming is more complicated. Nevertheless, in practice, a certain lack of repeatability is observed in the AMCM results, especially in the extreme percentiles of the _PDF_. A basic implementation of an adaptive Monte Carlo procedure involves carrying out an increasing number of Monte Carlo trials until the various results of interest have stabilized in a statistical sense. AMCM is based on going from an \(M\) sample to a larger one of size \(h\times M\) with a better statistical significance, being \(h\) a certain positive integer. The \(h\) value is obtained through an iterative process. Starting from the first sample of \(M\) size, more \(M\)-sized samples are generated independently at every iteration. The size samples which are being generated increase by \(2\times M\), \(3\times M\), \(4\times M\), \(\ldots\), \(h\times M\) until the statistical significance of \(h\times M\) sample is optimal compared with the numerical tolerance of the measurand typical uncertainty \(u(y)\). The optimal sample is used to estimate the measurand value, its typical uncertainty, and consequently the expanded uncertainty. In the MCM framework, the numerical tolerance \(\delta\) is the semi-width which delimits the last significant decimal of \(u(y)\), and oscillates according to the number of significant figures selected to express \(u(y)\). In this work, the number of significant figures is always two.