We address the problem of linear comparative calibration, a special case of linear calibration where both variables are measured with errors, and the analysis of the uncertainty of the measurement results obtained with the calibrated instrument. The concept is explained in detail using the calibration experiment of the pressure transducer and the subsequent analysis of the measurement uncertainties. In this context, the calibration and the measurements with the calibrated instrument are performed according to ISO Technical Specification 28037:2010 (here referred to as ISO linear calibration), based on the approximate linear calibration model and the application of the law of propagation of uncertainty (LPU) in this approximate model. Alternatively, estimates of the calibration line parameters, their standard uncertainties, the coverage intervals and the associated probability distributions are obtained using the Monte Carlo method (MCM) based on the law of propagation of distributions (LPD). Here we also obtain the probability distributions and the coverage interval for the quantities measured with the calibrated instrument. Furthermore, motivated by the model structure of this particular example, we conducted a simulation study that presents the empirical coverage probabilities of the ISO and MCM coverage intervals and investigates the influence of the sample size, i.e. the number of calibration points in the measurement range, and the different combinations of measurement uncertainties. The study generally confirms the good properties and validity of the ISO technical specification within the considered (limited) framework of experimental designs motivated by real-world application, with small uncertainties in relation to the measurement range. We also point out the potential weaknesses of this method that require increased user attention and emphasise the need for further research in this area.

中文翻译：

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ISO 线性校准和校准仪器获得结果的测量不确定度

我们解决了线性比较校准的问题，线性校准的一种特殊情况，其中两个变量的测量都有误差，并分析了使用校准仪器获得的测量结果的不确定性。使用压力传感器的校准实验和随后的测量不确定性分析详细解释了这个概念。在这种情况下，根据 ISO 技术规范 28037:2010（此处称为 ISO 线性校准），基于近似线性校准模型和不确定性传播定律的应用，使用被校准仪器进行校准和测量（LPU）在这个近似模型中。或者，校准线参数的估计，它们的标准不确定性，覆盖区间和相关的概率分布是使用基于分布传播定律 (LPD) 的蒙特卡罗方法 (MCM) 获得的。在这里，我们还获得了使用校准仪器测量的数量的概率分布和覆盖区间。此外，受此特定示例的模型结构的启发，我们进行了一项模拟研究，展示了 ISO 和 MCM 覆盖区间的经验覆盖概率，并研究了样本量的影响，即测量范围内的校准点数量，以及测量不确定度的不同组合。该研究总体上证实了 ISO 技术规范在考虑的（有限的）实验设计框架内的良好特性和有效性，这些框架是由实际应用驱动的，与测量范围相关的不确定性很小。我们还指出了这种方法的潜在弱点，需要更多用户关注，并强调需要在该领域进行进一步研究。