## 5 Numerical Simulation  
### Monte-Carlo Studies  
For this part, we carry out comprehensive Monte-Carlo simulations for the purpose of evaluating the various approaches for estimating the three parameters. For the point estimation, we adopt the mean squared error (MSE) and average bias (AB) since MSEis able to measure the volatility of data, and AB makes up for the disadvantage of MSE that it is sensitive to some extreme values. For the interval estimation, we select average interval length (AL) and the coverage probability (CP), as shorter ALs mean the confidence or credible intervals are more accurate under a fixed significance level, and higher CPs indicates that the confidence or credible intervals are more likely to include the true values of the parameters. The evaluating criteria for assessing the performance of the estimates outlined are shown below:  
1. Mean squared error (MSE) of the point estimation \(\hat{\phi}\), which is calculated by \(\frac{1}{N}\Sigma(\hat{\phi}-\phi)^{2}\), where \(\phi\) represents each of the three parameters discussed in this model. The number of replications is denoted by \(N\);
2. Average bias (AB) of the point estimation \(\hat{\phi}\), which is calculated by \(\frac{1}{N}\Sigma\mid\hat{\phi}-\phi\mid\);
3. Average length (AL) of the confidence or credible intervals (CIs) of \(\hat{\phi}\) with a significance level \(\tau\);
4. Coverage probability (CP) of the \(100(1-\tau)\%\) CIs of \(\phi\), which reflects the proportion of instances where the actual parameter values lie in the calculated intervals. To be concise, we take \(n_{1}=n_{2}=n\), \(m_{1}=m_{2}=m\) and \(r_{1}=r_{2}=r\), running simulations under both normal and accelerated conditions with the following CSs:  
1. \(n=25\), \(m=20\), \(r=5\), \(R_{j1}=\cdots=R_{j5}=1\), \(R_{j6}=\cdots=R_{j,15}=0\);
2. \(n=30\), \(m=20\), \(r=5\), \(R_{j1}=\cdots=R_{j5}=2\), \(R_{j6}=\cdots=R_{j,15}=0\);
3. \(n=25\), \(m=20\), \(r=5\), \(R_{j,11}=\cdots=R_{j,15}=1\), \(R_{j1}=\cdots=R_{j,10}=0\);
4. \(n=35\), \(m=30\), \(r=5\), \(R_{j1}=\cdots=R_{j5}=1\), \(R_{j6}=\cdots=R_{j,25}=0\);
5. \(n=40\), \(m=30\), \(r=5\), \(R_{j1}=\cdots=R_{j5}=2\), \(R_{j6}=\cdots=R_{j,25}=0\);
6. \(n=35\), \(m=30\), \(r=5\), \(R_{j,21}=\cdots=R_{j,25}=1\), \(R_{j1}=\cdots=R_{j,20}=0\);
7. \(n=55\), \(m=50\), \(r=5\), \(R_{j1}=\cdots=R_{j5}=1\), \(R_{j6}=\cdots=R_{j,45}=0\);
8. \(n=60\), \(m=50\), \(r=5\), \(R_{j1}=\cdots=R_{j5}=2\), \(R_{j6}=\cdots=R_{j,45}=0\);
9. \(n=55\), \(m=50\), \(r=5\), \(R_{j,41}=\cdots=R_{j,45}=1\), \(R_{j1}=\cdots=R_{j,40}=0\). Based on the above evaluating criteria and the censoring schemes, two sets of parameter values are taken into consideration for our Monte-Carlo studies, which are \(\mathbf{\theta_{1}}=(\alpha,\lambda,\beta)=(0.8,0.5,1.5)\) and \(\mathbf{\theta_{2}}=(\alpha,\lambda,\beta)=(1,0.8,2)\), respectively. In terms of the Bayesian estimate, the corresponding hyper-parameters are chosen as \((a_{1},b_{1},a_{2},b_{2},a_{3},b_{3})=(1.6,2,1,2,1,2)\) and \((a_{1},b_{1},a_{2},b_{2},a_{3},b_{3})=(4,4,3.2,4,4,4)\), which indicate the expectations of the three Gamma distributions as the prior, respectively. It makes sense that when choosing the more accurate priors, we may obtain better estimates. It is to be noted that the hyper-parameters should not be too large, since it may mask the effects of the three newly-constructed Gamma distributions, which led us to setting the hyper-parameters as above. The method in [15] is performed regarding the production of the general progressively censored sample under the pre-determined CS \((R_{r+1},R_{r+2},\cdots,R_{m})\), and the specific algorithms are shown below:  
Since this sort of censored sample can be obtained based on Algorithm 1, it is possible to generate sample under general progressive censoring by Algorithm 2.