# Risk Assessment for Linear Regression Models in Metrology
## 1 Introduction
For reasons that will be elaborated upon further in the text, it is not feasible to apply these simultaneous tolerance intervals for the model of risk assessment in regression that is presented in this paper. The tolerance interval in the metrology domain is established by the manufacturer in its specifications for a particular measuring device, or this interval is determined by the applicable standards. The risk assessment method outlined in this paper can be applied not only in metrology, but also in general in all regression models, regardless of the area in which the data originate. Therefore, this manuscript gives alternative values for the tolerance interval that can be used when the values for the tolerance interval provided by the manufacturer are unavailable. To reduce the impact of measurement uncertainty on risk assessment, an acceptance interval \([\mathrm{A_{L}},\mathrm{A_{U}}]\) is introduced into the procedure for evaluating the conformity of products with the prescribed specifications, in addition to the tolerance interval. The marks \(\mathrm{A_{L}}\) and \(\mathrm{A_{U}}\) represents the lower and upper limits of the acceptance interval, respectively. Acceptance and tolerance intervals can be in different mutual relations [24]. The limits of the tolerance interval and the acceptance interval depend on the properties of the measurand. These can be semi-open or semi-closed intervals [25]. Furthermore, they can be closed intervals, as in this case. If the tolerance interval is within the acceptance interval, it is a model of minimization of the producer's risk, as illustrated in Figure 1a. If the tolerance interval is outside the acceptance interval, it is a consumer risk minimization model (Figure 1c) [26]. A special situation when tolerance and acceptance interval overlap is when it is valid that \(\mathrm{A_{L}}=\mathrm{T_{L}}\) and \(\mathrm{A_{U}}=\mathrm{T_{U}}\). This is called shared risk (Figure 1b). The guard band of length \(w\), between the tolerance interval and the acceptance interval, ensures a reduction in the probability of making wrong decisions. In practice, it is commonly recommended to minimize the consumer's risk to enhance the quality of the delivered products. However, it is feasible to set the length of the guard band so that both the customer and the producer are satisfied. When measuring, it is assumed that the object of interest has a measurable property \(Y\) with possible values \(\eta\). It is natural to assume that there exists historical data about the item of interest, such as those found in manuals, scientific papers, data from previous measurements, etc. Also, one can talk about prior beliefs of the measurer based on experience regarding possible values that will be obtained during the measurement, or regarding the possible distribution of parameters that describe the measured data. Such data are treated as random variables. Knowledge about these data is given by prior distribution, that is, by the probability density function (PDF), designated with \(g_{0}(\eta)\)[13]. Depending on the available data, the prior can be a non-parametric, one-parameter, or two-parameter distribution [27; 28]. According to the principle of maximum entropy (PME), if data for two moments are available, a two-parameter distribution is used for the prior [29; 30; 31]. In this manuscript, the prior is a normal distribution whose parameters are the best estimate of a measurand \(y_{0}\), with an associated standard measurement uncertainty \(y_{m}\). The data associated with the measurements \(Y_{m}\) are also treated as a random variable. The values that this random variable can take are denoted by \(\eta_{\text{m}}\). Random variable \(Y_{m}\) is modeled via the likelihood function for normal distribution and denoted by \(\text{h}(\eta_{\text{m}}|\eta)\)[13]. The formula for the likelihood function involves standard measurement uncertainty \(\text{u}_{\text{m}}\) of some future measurement.