## 4 Numerical Examples
### Case Application and Sensitivity Analysis  
TWT has the merit of high reliability, long lifetime, high gain, and large power which is one of the most extensively used vacuum electronic devices [27], thus it is widely applied in the area of aerospace. TWT mainly consists of an electronic gun, slow-wave structure, RF input and output, permanent magnet and collector [28]. The cathode is an important part of the electron gun, which determines the life of TWTs under good vacuum conditions [29], and it is called the heart of TWT. The CSADT for TWTs is carried out under three temperature levels: 1020 \({}^{\circ}\)C, 1060 \({}^{\circ}\)C, and 1100 \({}^{\circ}\)C. The normal operational level is 980 \({}^{\circ}\)C. Five cathodes are put to the test at three stress levels and the degradation data are recorded every 1000 h until 77,000 h. The emission current of a cathode decreases over time, causing a soft failure when its emission current drops below a critical failure threshold. The performance degradation \(\Delta y\) is the change of cathode emission current \(y\) relative to the initial value \(y_{0}\) at a certain time. However, the cathode emission current decreases gradually with time, for the sake of ensuring that the degradation process of the cathode is a positive drift, that is, the drift parameter is positive, the performance degradation takes the decreased value of the cathode emission current at a certain moment relative to \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \(\hat{\mu}_{\mathbf{q}}\) & \(\hat{\sigma}_{\mathbf{q}}\) & \(\hat{\mathbf{b}}\) & \(\kappa\) & \(\hat{\sigma}_{\mathbf{B}}\) & \(\hat{\sigma}_{\mathbf{\epsilon}}\) & \(\ell\) & \(\mathbf{AIC}\) \\ \hline \(M_{0}\) & 1.0223 & 0.2915 & 1934.6 & 0.032 & / & 0.036 & 4511.7 & \(-\)9011.4 \\ \hline \(M_{1}\) & 1.2889 & 0.3120 & 1831.7 & / & 0.0071 & 0.027 & 4241.6 & \(-\)8473.2 \\ \hline \(M_{2}\) & 1.3781 & 0.0001 & 1853.1 & / & 0.026 & / & 3691.2 & \(-\)7374.4 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparisons of three different degradation models with LEDs CSADT degradation data. Figure 1: The degradation trajectory of LEDs under CSADT over time t at (**a**) 25 \({}^{\circ}\)C, (**b**) 65 \({}^{\circ}\)C, (**c**) 105 \({}^{\circ}\)C.  
the initial value. That is, \(\Delta\text{y}=y_{0}-y\). Therefore, the failure threshold is positive. The cathode is considered to have failed when the relative degradation value of the emission current exceeds 10% of the initial emission current whose failure threshold is \(D=13\)[30]. The degradation paths of the fifteen tested units under three stress levels are described in Figure 2, showing that the degradation of samples uniformly exhibits a nonlinear characteristic especially at the beginning of the CSADT. Thus, the degradation process is modeled with a time-scale transformed Wiener process as Equation (17) with \(\Lambda(t)=t^{\theta}\), which lets \(\theta=0.5\) for simplicity according to the degradation trajectory. According to [29], the degradation rate of cathodes at a constant emission current can be described by the Arrhenius model. Therefore, we use a stochastic Arrhenius model to represent the acceleration model applied to determine the relationship between the failure time data and the working temperature stress. Based on the parameters estimation method proposed in Section 3, we can obtain the estimation results of parameters and the \(AIC\) as shown in Table 3, indicating that \(M_{0}\) clearly outperforms other models in terms of \(AIC\). It is shown that the proposed model has a better fit than the others. Figure 2: Degradation trajectory of cathode emission current at (**a**) 1020 \({}^{\circ}\)C, (**b**) 1060 \({}^{\circ}\)C, (**c**) 1100 \({}^{\circ}\)C.  
In Figure 3, we describe the PDF and CDF under the normal operating temperature \(T_{0}=980^{\circ}\)C based on the parameter estimation results presented in Table 3. The time corresponding to the peak values of PDF were as follows: 0.0055, 0.0038, and 0.002 for \(M_{0}\) to \(M_{2}\), respectively.