## 6 Case Application  
According to the analysis in Section 2, the performance of space TWTA directly affects the stable operation and reliability of the system. As important telemetry parameters of TWTA, the helix current and anode voltage gradually deteriorate with the extension of in-orbit time, whose relative failure thresholds are 2.5 mA and 400 V, respectively, representing the health status of TWTA, and provide effective information for the reliable life prediction of satellite subsystems. In the early phase of TWTA in-orbit, the product was still aging, and its performance was not very stable, and the performance parameters of the three TWTAs were recorded according to the remote monitoring system in this paper. All data were initialized and preprocessed. The parameter degradation trajectories after aging are shown in Figure 3, which shows the performance degradation trajectory of individual samples of TWTA, the changes of the average helix current and anode voltage of the in-orbit products, and the uncertainty brought by the linear model fitting. Figure 3: The degradation trajectory of TWTA. (**a**) Helix current degradation path and (**b**) anode voltage degradation path. To determine whether the degradation paths of the three TWTA's performance parameters follow the Wiener process and Gamma process separately, a goodness-of-fit was performed on the degradation data of the helix current and anode voltage. The Kolmogorov-Smirnov (K-S) test was used to test the normal distribution and Gamma distribution of goodness-of-fit for the degradation increments of the three samples with a 95% confidence level. The results of the K-S test for all samples were greater than 5% and show that it is capable of degradation modeling of the Wiener process and the Gamma process. According to the model built in Section 3 and the MCMC parameter estimation method, sampling and estimating unknown parameters were carried out based on the Wiener process and the Gamma process, and all unknown parameters were assumed to have no prior distribution. The estimation results of unknown parameters are shown in Table  
Due to the few in-orbit samples, the overall features, such as \(\sigma^{2}\) and \(\alpha\), reflect-ing the performance degradation of the TWTA, showed more volatility in the itera-tions, and resulted in a larger estimation error than the other parameters. Therefore, for the above problems, the parameters reflecting the overall characteristics of the product were resampled and re-estimated using the Bayes-Bootstrap method. The re-sults after following this procedure are shown in Table 3.  
The evaluation results of marginal distributions were substituted into Equation (24), and the correlation coefficient \(\theta\) was obtained by MLE, which is shown in Table 4. Through the comprehensive determination of the Copula function, the AIC values of different Copula functions, including the Frank, Clayton, and Gumbel functions, are shown in  
Figure 5: The parameter posterior distribution of anode voltage (**a**) \(\alpha\), (**b**) \(\kappa\), (**c**) \(r\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Parameter** & **Evaluation** \\ \hline \multirow{2}{*}{MCMC} & \(\hat{\sigma^{2}}\) & 0.044 \\  & \(\hat{\alpha}\) & 0.983 \\ \hline \multirow{2}{*}{Bootstrap} & \(\hat{\sigma^{2}}\) & 0.04 \\  & \(\hat{\alpha}\) & 1.04 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The estimation results of unknown parameters. From Figure 6, we can find the following phenomena. The reliability results of the four models are different, and the reliability of a single-parameter degradation model is higher than the reliability of the model proposed in this paper. With the increase of time, the difference among the models also increased.