## 5 Case Study
### Comparison of the First-Passage Time Distributions
Critical values should be defined based on historical behavior, but a first approach can be considered as follows: consider that the measurement system has been evaluated and an estimation of \(\sigma\) is obtained, then the true and observed first-passage time distributions can be characterized. Consider that \(\sigma\) is a component of variance with good performance, i.e., less than 10% of the total variation of the process [56]. If the mean is the estimation of interest, then from (14) it can be considered that \(C_{E}=I_{E}\), this equivalence of the critical value will relate the good performance of the measurement system to the estimation of the mean failure time, so the control scheme can be initiated. Now consider a specific example, if the critical values are considered as \(C_{CV}=0.66,C_{Var}=0.2,C_{E}=0.12\), and \(C_{(z_{5})}=0.24\), it is observed from Table 5 that the standard deviation for the measurement error \(\sigma=0.0006058\) is good enough for the performance of the measurement system. With this parameter of the measurement error, it is expected that the estimated lifetime from the data contaminated with measurement error should be accurate enough. It should be noted that \(\sigma\) may be optimized by following the sequence determined in the flow chart in Figure 2. However, previous knowledge of the case study must be available such that optimal values of the critical values \((C_{CV},C_{var},C_{E},C_{z_{1}})\) are defined. For instance, the mean time to failure (MTTF) of the observed degradation and the true degradation with \(\sigma=0.0006058\) are \(E(T_{o})=1.088\), and \(E(T_{s})=1.211\), respectively. Which means a difference of 0.123 hundred thousands of cycles. If the maximum allowance for the difference between MTTF is expected to be, for example, 0.05 hundred thousands of cycles and the critical value \(C_{E}=0.043\), it can be noted that \(I_{E}(T_{s},T_{o})\) is higher \(C_{E}\) which means that the measurement system should be improved such that the measurement process is executed more accurately and the variation caused by the measurement system is reduced. The same approach may be considered for any other index different than \(C_{E}\), depending on the estimation of interest. For example, if it is expected that the differences between the 5 th percentiles of the failure times for the observed degradation and the true degradation be 0.09 hundred thousands of cycles and \(C_{z_{5}}=0.105\), again it can be noted that \(I_{z_{5}}>C_{z_{5}}\), which denotes a high variance of the measurement error. The reliability functions with and without measurement error were obtained based on the corresponding first-passage time distributions. The respective differences can be noted in Figure 8. Along with the respective reliability functions, we also present the Kaplan-Meier reliability for the observed and true degradation, along with their respective 95% confidence intervals. At different \(t_{j}\), the Kaplan-Meier confidence interval of the true reliability does not include the observed reliability, which denotes the difference. A difference of the reliability functions presented by [PERSON] et al. [30], apart from the considered stochastic processes for the observed degradation, relies on that, in this paper, a stochastic process is fitted to the true degradation which defines the dashed red reliability function. Furthermore, the reliability estimation from Kaplan-Meier results in a different behavior as the true degradation comes from a gamma-Gaussian deconvolution. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \(I_{CV}(T_{s},T_{o})\) & \(I_{Var}(T_{s},T_{o})\) & \(I_{E}(T_{s},T_{o})\) & \(I_{z_{5}}(T_{s},T_{o})\) \\ \hline Index & 0.6574 & 0.1919 & 0.1136 & 0.2309 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Computed indices from the observed and true first-passage time distributions.