## 5 Case Study
### Comparison of the First-Passage Time Distributions  
It is considered that the vector \(W[k]\) is described by a gamma process as \(Ga(v^{*}\Delta t_{j},u^{*})\) with shape parameter \((v^{*}\Delta t_{j})\), and scale parameter \((u^{*})\). The true first-passage time distribution can be obtained by considering the Birnbaum-Saunders distribution with parameters \(\alpha_{s}^{*}=\sqrt{u^{*}/(\omega-z_{0})}\) and \(\beta_{s}^{*}=(\omega-z_{0})/v^{*}u^{*}\), with \(z_{0}=0\), and \(\omega=0.4\). Thus, the CDF is described as  
\[F_{T_{s}}(t)=\Phi\Bigg{[}\frac{1}{\alpha_{s}^{*}}\Bigg{(}\sqrt{\frac{t}{\beta _{s}^{*}}}-\sqrt{\frac{\beta_{s}^{*}}{t}}\Bigg{)}\Bigg{]}, \tag{16}\]  
with mean obtained as \(E(T_{s})=\beta_{s}^{*}\big{(}1+\alpha_{s}^{*2}/2\big{)}\), and the variance as \(Var(T_{s})=\alpha_{s}^{*}\beta_{s}^{*}\)\(\big{(}1+5\alpha_{s}^{*2}/4\big{)}^{1/2}\). The estimated gamma parameters of the true degradation were obtained as \(\vartheta^{*}=37.9237\) and \(\hat{u}^{*}=0.0088\). Considering these estimates, the parameters of the first-passage time distributions for the true degradation can be easily obtained considering the Birnbaum-Saunders distribution. The computed parameters were obtained as \(\hat{\alpha}_{s}^{*}=0.1483\) and \(\hat{\beta}_{s}^{*}=1.1985\). In Table 4, a comparison of the the mean, variance, and CV for the observed and true first-passage time distributions is presented. From Table 4, the effect of the measurement over the first-passage time distribution is reflected. For instance, the mean passage-time from the true degradation is greater that the obtained from the observed degradation. In addition, the variance is smaller for the true degradation compared to the observed degradation. This finding can be confirmed by the degradation paths described in Figure 6, where the mean degradation and the variation among degradation paths are smaller compared to the observed degradation. The distributions \(f_{T_{e}}(t)\) and \(f_{T_{s}}(t)\) are compared by computing the indices described in ((12)-(15)) as denoted in the flow chart in Figure 2. The 5 th percentile is considered for (15). The quantile function of the Birnbaum-Saunders distribution is described as [55],  
\[t(q)=\beta\Bigg{(}\frac{\alpha z_{q}}{2}+\sqrt{\frac{\alpha^{2}z_{q}^{2}}{4}+ 1}\Bigg{)}^{2},\]  
where \(z_{q}\) is the \(q\times 100\)th quantile of the standard normal distribution. Considering that \(z_{5}=-1.6448\) and the estimates \(\hat{\alpha}_{\sigma}^{*}\), \(\hat{\beta}_{\sigma}^{*}\) and \(\hat{\alpha}_{s}^{*}\), \(\hat{\beta}_{s}^{*}\), the 5 th percentile for the first-passage time distributions for the observed and true degradation were obtained as \(t_{o}(5)=0.7633\) and \(t_{s}(5)=0.9396\), respectively. All four indices described in Section 4, \(I_{CV}(T_{s},T_{o})\), \(I_{Var}(T_{s},T_{o})\), \(I_{E}(T_{s},T_{o})\), \(I_{z_{5}}(T_{s},T_{o})\) were computed and are presented in Table 5. \begin{table}
\begin{tabular}{l l l l} \hline \hline  & **Mean** & **Variance** & **CV** \\ \hline Observed & 1.088 & 0.223 & 0.434 \\ True & 1.211 & 0.18 & 0.1487 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the mean, variance and CV for the first-passage times of the observed and true degradation. The critical values \((C_{CV},C_{var},C_{E},C_{z_{5}})\) may be defined depending of the allowance for the measurement error of the measurement system. In this paper, four indices are considered, however, one index can be used depending of the reliability estimation of interest. As can be noted from ((12)-(15)), the indices are ratios that account for the relative increase in each reliability estimation, i.e., CV, variance, mean, percentiles. Indeed, the greater the value of the respective indices the more difference between reliability estimations, which means more variability of the measurement error, i.e., \(\sigma\).