## 5 Numerical Simulation
### Real Data Examples  
For this part, we examine a real lifetime dataset that records a series of oil breakdown times of a kind of insulating fluid under different stress levels (voltages, KV) as documented by [23], Chapter 3, Table 3.1, to validate the previously proposed methods. To determine whether the GIED could be an appropriate lifetime distribution in terms of this real dataset, we first derive the AIC (Akaike information criterion) of several commonly utilized lifetime distributions. The results for AICs are presented in Table 3.  
Figure 5: Plots of evaluating criteria for \(\theta_{2}\).  
From the table results, we can conclude that GIED provides a better goodness of fit over the other frequently used models since it has the smallest AIC.  
To further demonstrate the appropriateness of the GIED, the Kolmogorov-Smirnov (K-S) test is utilized. The specific samples under two levels of stress (32 KV as normal condition and 36 KV as accelerated condition) is offered in Table 4. According to the results in Table 3, the data are assumed to follow GIED, thus one can further conduct the K-S test:  
\[\mathrm{H_{0}}:\text{ the samples are from GIED \ \ \ \ }vs.\ \ \ \mathrm{H_{1}}:\text{ the samples are not from GIED,}\]  
then the corresponding K-S distances as well as \(p\)-values are calculated and shown in Table 4 as well. When the \(p\)-values are larger than the significance level, one have good reason to accept the null hypothesis, which means determining the samples are from a GIED. Under normal and the accelerated condition, the calculated \(p\)-values are 0.4149 and 0.7867 respectively, which means that one should accept the null hypothesis \(\mathrm{H_{0}}\), leading to a fundamental conclusion that GIED is well-fitting to this dataset. In addition, the empirical cumulative distribution plots of both normal (32 KV) and accelerated (36 KV) condition with the fitted CDF curves where the unknown parameters are given by MLEs are shown in Figure 6, which can be a further illustration of the validity of using GIED model.  
Based on Table 4, we generate two new sets of general progressively Type-II censored samples using a specified CS where \(n=15,m=12,r=2\) and \(R_{1}=R_{2}=R_{3}=1,R_{4}=\dots=R_{10}=0\). The new data sets are shown in Table 5. Hence, one can conduct the classical and Bayesian methods mentioned above on the new data set to derive point and interval estimation of \(\alpha\), \(\lambda\) as well as \(\beta\). For Bayesian methods, because there is no available extra informative knowledge about the three parameters, we set all the hyper-parameters as 0.001, which indicates that the pre-determined priors are nearly non-informative. Tables 6 and 7 display the outcomes of different methods, which clearly demonstrate that the effectiveness of MLEs and BEs is highly comparable since they have produced similar estimates of the parameters, whereas for the interval estimation, BCIs works better than ACIs featuring much shorter interval lengths.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Stress** & \multicolumn{4}{c}{**Failure Times**} & \multicolumn{4}{c}{**K-S Distance**} & \(p\)**-Value** \\ \hline
32 KV & 0.27 & 0.40 & 0.69 & 0.79 & 2.75 & 0.2180 & 0.4149 \\  & 3.91 & 9.88 & 13.95 & 15.93 & 27.80 & & \\  & 53.24 & 82.85 & 89.29 &  
Through an analysis of the practical oil breakdown times data set, one can conclude that both classical and Bayesian method are helpful for estimating the parameters of GIED by utilizing general progressively censored samples in a CSPALT, and additionally, Bayesian estimates performed better in terms of the interval lengths.