## 5 Case Study  
The dataset presented by [PERSON] et al. [52] is considered for the implementation of the proposed approach. This case study consists in the crack-growth of a terminal in an electronic device. The function of this terminal is to transfer a signal to a receptor, which can be disrupted if the crack in the terminal propagates to a certain critical level, and thus would lead to a failure of the device. A DT was carried out to study the propagation of the crack in 10 terminals. The crack propagation was measured every 0.1 hundred thousand cycles until 0.9 hundred thousand cycles. In this article, it is considered that a failure is said to have occurred when the length of the crack exceeds the critical length of 0.4 mm. The total of sample devices are \(N=10\), with \(M=9\) observation times as \(j=1,2,3,4,5,6,7,8,9\), which are the same for all the \(i=1,2,\ldots,10\) samples with \(t_{j}=(0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9)\) hundred thousand cycles. In Table 1, the degradation measurements are presented, the units are millimeters. In Figure 3, the cumulative degradation paths are presented.  
It is assumed that the degradation data in Table 1 are governed by a gamma process as in (4). Thus, \(Z_{i}(t_{j})\) is the observed degradation for \(i=1,2,\ldots,10\), and  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Device**} & \multicolumn{8}{c}{**Hundred Thousands of Cycles**} \\ \cline{2-11}  & **0** & **0.1** & **0.2** & **0.3** & **0.4** & **0.5** & **0.6** & **0.7** & **0.8** & **0.9** \\ \hline
1 & 0 & 0.01 & 0.03 & 0.055 & 0.107 & 0.165 & 0.183 & 0.2 & 0.26 & 0.302 \\
2 & 0 & 0.09 & 0.161 & 0.172 & 0.247 & 0.259 & 0.281 & 0.371 & 0.401 & 0.429 \\
3 & 0 & 0.01 & 0.06 & 0.081 & 0.118 & 0.142 & 0.158 & 0.169 & 0.232 & 0.262 \\
4 & 0 & 0.016 & 0.076 & 0.087 & 0.104 & 0.127 & 0.198 & 0.208 & 0.218 & 0.258 \\
5 & 0 & 0.036 & 0.096 & 0.176 & 0.204 & 0.242 & 0.281 & 0.325 & 0.415 & 0.495 \\
6 & 0 & 0.014 & 0.102 & 0.112 & 0.194 & 0.277 & 0.289 & 0.305 & 0.335 & 0.391 \\
7 & 0 & 0.037 & 0.064 & 0.078 & 0.096 & 0.124 & 0.164 & 0.234 & 0.254 & 0.326 \\
8 & 0 & 0.035 & 0.086 & 0.105 & 0.174 & 0.267 & 0.277 & 0.347 & 0.361 & 0.384 \\
9 & 0 & 0.067 & 0.148 & 0.161 & 0.173 & 0.184 & 0.218 & 0.229 & 0.239 & 0.285 \\
10 & 0 & 0.025 & 0.052 & 0.064 & 0.076 & 0.151 & 0.187 & 0.205 & 0.222 & 0.262 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Degradation dataset of crack-growth case study.  
Figure 3: Cumulative degradation paths of the case study.  
\(t_{j}=(0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9)\). In this case study, it is considered that the measurement error is described by a Gaussian distribution with \(\mu=0\) and \(\sigma\), as described in (5). Thus, by considering the flow chart in Figure 2, we first estimate the parameters of the gamma process for the degradation dataset in Table 1, then we estimate \(\sigma\) by performing an R&R study to the measurement system. Next, we illustrate the effect of the measurement error over the true degradation distribution by using the deconvolution modeling proposed in (9), and to assess the effect over the free-error first-passage time distribution by using the indices presented in ((12)-(15)).