## 5 Proposals for AMCM Improvement  
The selection criteria for the \(h\) value in GUM-S1 assumes that the mean values of the statistical descriptors \(y\), \(u(y)\), \(y_{low}\), \(y_{high}\) at every iteration follow a normal distribution and therefore, such selection criterion would have a 95.45% confidence level. Strictly speaking, this is correct when \(h\) is large enough, but in practice, this is not so. This implies that the factor multiplying \(s(\overline{y})\), \(s[\overline{u}(y)]\), \(s(\overline{y}_{low})\), \(s\left(\overline{y}_{high}\right)\) is underestimated. In this work, the first proposal is to use a coverage factor from a Student's t-distribution with \(h-1\) degrees of freedom. In the case that the number of degrees of freedom is large enough and compatible with Gaussian statistics, the Student's t-distribution itself will be very similar to a Gaussian and the GUM-S1 criteria could be used. In principle, the Student's t-distribution coverage factor which should be used would be the one corresponding to a 95.45% confidence level, as in GUM-S1 the standard deviations \(s(\overline{y})\), \(s[\overline{u}(y)]\), \(s(\overline{y}_{low})\), \(s\left(\overline{y}_{high}\right)\) are multiplied by 2. A higher confidence level could be used (for instance 99.73%) but this would imply more iterations to converge \(h\). It is proposed to use the t-Student factor instead of the z factor of the normal distribution used by GUM-S1 to correct the misuse of applying the \(z\) factor of the normal distribution to a case of small samples. That is, it is proposed to correctly treat the small samples obtained with the first iterations of the method (for example, when there are two or three elements at the beginning). As is well known, when the samples increase in size, the \(z\) factor is adequate. But it is also well known that when the samples increase in size, t-Student tends to \(z\). Therefore, the first improvement proposal for AMCM consists of changing the acceptance and rejection criterion for \(h\) from GUM-S1 so that this criterion is also adaptive; that is, the stricter the criterion the lower the \(h\) value. The level of confidence in the optimization of \(h\) would be 95.45% (GUM-S1) or higher depending on the requirements in each case. As a second improvement, this work proposes the validation of the AMCM repeatability by using \(\delta\) as a reference. In practice, when a new cycle of iterations is performed, the lack of repeatability could lead to the variability of \(y\), \(u(y)\), \(y_{low}\), \(y_{high}\) and a new value of \(h\), \(h\
eq h^{\prime}\) would be obtained. Therefore, the proposal considers \(R\) repetitions of the adaptive cycle and checks if \(y\), \(u(y)\), \(y_{low}\), \(y_{high}\) are changing if rounded according to \(\delta\). This rounding process is conducted first in \(u(y)\) according to \(\delta\) and then in \(y\), \(y_{low}\), \(y_{high}\) according to \(u(y)\). The precision of the final results of AMCM must be always adjusted to \(\delta\); otherwise, a spurious variability would always be observed. In order to come up with a stable result, the \(R\) values for \(y\), \(u(y)\), \(y_{low}\), \(y_{high}\) will be averaged and rounded according to \(\delta\). The stability will be confirmed by calculating the standard deviations of the sample means multiplied by 2 or 3, depending on the requirements, and the results will be checked to be zero after rounded according to \(\delta\). This will ensure that the repeatability of the results \(y\), \(u(y)\), \(y_{low}\), \(y_{high}\) will have a 95.45% or 99.73% confidence level whenever \(R\) is large enough. In this work, \(R\) is taken as 50. The explanation for taking 50 iterations of the adaptive cycle is as follows. When repeating the application of the AMCM, the authors have found that the results fluctuate, which provides a certain lack of definition to the result.