## 2 Modeling of the Observed Degradation via Gamma Process  
In this article, it is considered that the degradation measurements of a certain performance characteristic are contaminated with measurement error. These measurements are considered as the observed degradation, as these are directly observed. The modeling of this characteristic is firstly discussed in this section. In this case, stochastic modeling of the degradation process is considered given that it is possible to introduce the temporal uncertainty of the degradation increments over time [40]. The gamma process is specifically considered to describe the observed degradation of a characteristic of interest. This process has been widely documented and implemented in multiple case studies in the literature [40; 41; 42; 43], this given its characteristics that it is a monotone stochastic process with independent and non-negative increments. We consider \(\{Z(t),t\geq 0\}\) as a degradation process that describes the observed degradation of a performance characteristic over time, it is deemed that \(Z(t)\) is governed by a gamma process with the following properties: the degradation increments \(Z(t+\Delta t)-Z(t)=\Delta Z(t)\) follow a gamma distribution \(Ga(v[t+\Delta t-t],u)\), and \(\Delta Z(t)\) are independent \(\forall t_{1}<t_{2}<t_{3}<t_{4}\). From the PDF of the gamma process, the parameter \(vt\) is a non-negative shape parameter with \(t\geq 0\), \(v(0)\equiv 0\), while \(u>0\) is the scale parameter. It is known that the mean and variance of the processes are defined as \(vt\cdot u\) and \(vt\cdot u^{2}\), respectively. Thus, \(\Delta Z(t)\) has the following PDF,  
\[f(\Delta Z(t)|v,u)=\frac{\Delta Z(t)^{vt-1}}{u^{vt}\Gamma(vt)}\,exp\left\{- \frac{\Delta Z(t)}{u}\right\}. \tag{1}\]An important aspect of the reliability assessment of degradation processes is related to the first-passage times, these are events described by the moment when the cumulative degradation reaches a critical level \(\omega\). Thus, the first-passage time of the observed degradation is defined as \(T_{o}=inf\{t_{o}:Z(t)\geq\omega\}\). The cumulative degradation can be used to describe the cumulative distribution function (CDF) of the first-passage times as \(P(Z(t)\geq\omega)=1-F_{Ga}(\omega,\upsilon,u)\), which results as,  
\[P(Z(t)\geq\omega)=\int_{\omega}^{\infty}f_{Z(t)}(z)dz=\frac{\Gamma(\upsilon t,\omega/u)}{\Gamma(\upsilon t)}. \tag{2}\]  
The first-passage time CDF in (2) can be related to the Birnbaum-Saunders distribution [44; 45], with parameters \(\alpha_{o}^{*}=\sqrt{u/(\omega-z_{0})}\) and \(\beta_{o}^{*}=(\omega-z_{0})/\upsilon w\), where \(z_{0}\) is the initial level of degradation. The CDF is defined as follows  
\[F_{T_{a}}(t)=\Phi\Bigg{[}\frac{1}{\alpha_{o}^{*}}\Bigg{(}\sqrt{\frac{t}{\beta _{o}^{*}}}-\sqrt{\frac{\beta_{o}^{*}}{t}}\Bigg{)}\Bigg{]}, \tag{3}\]  
where \(\Phi\) denotes the standard normal CDF. The mean of the first-passage time distribution is obtained as \(E(T_{o})=\beta_{o}^{*}\big{(}1+\alpha_{o}^{*2}/2\big{)}\), and the variance as \(Var(T_{o})=\alpha_{o}^{*}\beta_{o}^{*}\big{(}1+5\alpha_{o}^{*2}/4\big{)}^{1/2}\). As (1) denotes the PDF of the observed degradation, let us consider a scheme of a degradation test where \(i=1,2,\ldots,N\) units are tested and \(j=1,2,\ldots,M\) denotes the total number of measurements for all the tested units, which results in observed degradation measurements \(Z_{i}(t_{j})\). Then, it is defined that the degradation increments \(\Delta Z_{i}(t_{j})=Z_{i}(t_{j})-Z_{i}\Big{(}t_{(j-1)}\Big{)}\), with \(t_{0}=0\), \(\Delta t_{j}=t_{j}-t_{(j-1)}\), have the next PDF,  
\[f\big{(}\Delta Z_{i}(t_{j})|\upsilon,u\big{)}=\frac{\Delta Z_{i}(t_{j})^{ \alpha t_{j}-1}}{u^{\alpha t_{j}}\Gamma\big{(}\upsilon\Delta t_{j}\big{)}} exp\Bigg{\{}-\frac{\Delta Z_{i}(t_{j})}{u}\Bigg{\}}. \tag{4}\]  
As mentioned earlier, normally the observed degradation is contaminated with measurement error. Which implies that the true degradation cannot be observed directly from the degradation process.