## 5 Numerical Simulation
### Monte-Carlo Studies
Different estimates adopting both classical and Bayesian viewpoint for the three parameters and the corresponding evaluating criteria (EC) are calculated with sample generated from both normal and accelerated conditions utilizing general progressively Type-II censoring under multiple CSs. Specifically, the MSEs and ABs of the point estimation are shown in Table 1, and the ALs and CPs of the interval estimation are given in Table 2. In addition, the related plots of the criteria for different parameters are presented in Figures 4 and 5 as well. ```
Step 1: Generate \(m\) Uniform\((0,1)\) random variables \(W_{1},W_{2},\cdots,W_{m}\) as observations; Step 2: Let \(V_{i}=W_{i}^{1/\left(i+\sum_{j=n-i+1}^{m}R_{j}\right)},i=1,2,\cdots,m\). Step 3: Let \(U_{i}=1-\prod_{j=m-i+1}^{m}V_{j}\), \(i=1,2,\cdots,m\). Consequently, \(U_{i}\)'s form the progressively Type-II right censored sample of Uniform\((0,1)\). Step 4: Let \(X_{i}=F^{-1}(U_{i})\) for each \(i\). \(F^{-1}(\cdot)\) denotes the inverse CDF corresponding to the lifetime distribution being studied. Consequently, \(X_{1},\cdots,X_{m}\) constitute the necessary progressively Type-II censored sample of \(F(\cdot)\). ```  
**Algorithm 1**Generating Progressively Type-II Censored Samples  
```
Step 1: Generate \(U_{r+1}\) from Beta\((r+1,n-r)\); Step 2: By adopting Algorithm 1, generate \(m-r-1\) progressively Type-II censored \(\text{Uniform}(0,1)\) samples using CS \((R_{r+2},\cdots,R_{m})\), and they are denoted by \(U_{(r+2)},\cdots,U_{(m)}\); Step 3: Let \(U_{i}=U_{(i)}(1-U_{r+1})+U_{r+1}\), \(i=r+2,\cdots,m\), in which \(U_{(i)}\) is as derived according to Step 2; Step 4: Let \(X_{i}=F^{-1}(U_{i})\), for each \(i\). Consequently, \(X_{r+1},\cdots,X_{m}\) constitute a general progressively Type-II censored sample of \(F(\cdot)\). ```  
**Algorithm 2**Generating General Progressively Type-II Censored Samples  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & & & \multicolumn{4}{c}{\(\theta_{1}\)} & \multicolumn{4}{c}{\(\theta_{2}\)} \\ \hline  & & & \multicolumn{4}{c}{**MLE**} & \multicolumn{4}{c}{**BE**} & \multicolumn{4}{c}{**MLE**} &  
The findings of the simulation, which are shown in the tables and figures, allow for the formation of several different conclusions:  
1. In most of the cases, for point estimation and interval estimation, Bayesian method performs better than the classical estimation, with smaller MSEs and ABs, shorter ALs and higher CPs. For example, in terms of the parameter \(\alpha\) of \(\mathbf{\theta_{1}}\) under CS 7-9, one can find from Table 1 that the MSEs for MLE lie around 0.0210 whereas for BE is around 0.0160, with the ALs being about 0.4480 and 0.3739 respectively, which can be seen as a significant advantage of Bayesian method. 2. For fixed CSs, both point and interval estimation see a considerable progress when the number of valid samples accumulates. One can see from the two tables that when 10 samples are censored, CS 8 providing \(m=50\) valid failure times has much smaller MSEs (0.0210, 0.0160) and ALs (0.4485, 0.3728) than CS 5 and CS 2 (considering a in \(\mathbf{\theta_{1}}\), for other parameters we can also obtain similar findings), which indicates the consistency of the corresponding estimation. 3. However, the performances of both classical and Bayesian method are not satisfying with small samples, providing relatively large MSEs (for instance, around 0.25 for MLE of \(\beta\) in \(\mathbf{\theta_{1}}\)) and long intervals (over 1.6 for ACI of \(\beta\) in \(\mathbf{\theta_{1}}\)). BEs and BCIs using small samples are slightly better. It is to be noted that Bayesian method has a distinct advantage over the classical method in the small sample cases, where BEs have much smaller MSEs and ABs and BCIs give much shorter intervals together with relatively higher CPs.