## 5 Simulation and Data Analysis
### Simulation Study
As shown in Table 1, reducing \(\epsilon\) below 0.0001 does not significantly improve the MSE of parameter estimates. Thus, \(\epsilon=0.0001\) is chosen as the optimal threshold. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\epsilon\) & \(a_{1}\) & \(a_{2}\) & \(b\) \\ \hline
0.001 & 0.01697/7448 & 0.034834531 & 0.016680585 \\
0.0001 & 0.016734582 & 0.035520978 & 0.016437775 \\
0.00001 & 0.016355864 & 0.035480364 & 0.016265085 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: MSEs of the MLE obtained by EM algorithm with different values of \(\epsilon\). Tables 2-4 show the results of the point estimators, including the MLEs and Bayesian estimators for \(a_{1}\), \(a_{2}\), and \(b\), where '\(OPT\)' and '\(EM\)', respectively, refer to the MLE results obtained by the _optim_ function and the EM algorithm. And prior 1 refers to the non-informative prior, while prior 2 refers to the informative prior. In addition, pc refers to the censoring proportion under the corresponding scheme and '\(BE\)' refers to the Bayesian estimator. We discuss different censoring schemes by setting different values of the parameters \(\alpha\) and \(\beta\). According to Tables 2-4, several conclusions are drawn as follows:  
(1) Under different values of \((\alpha,\beta)\), which represent different censoring schemes, the biases and MSEs of each estimator are very similar. This fully demonstrates that the model is robust to middle-censored data. (2) As anticipated, for every estimator, both the bias and the MSE decline as the sample size \(n\) becomes larger. This indicates an improvement in their performance. Figure 8: Histograms of Gibbs samples for parameters and censored lifetimes when \(n=100\) and \((\alpha,\beta)=(0.5,0.5)\). Figure 6: Histograms of Gibbs samples for parameters and censored lifetimes when \(n=30\) and \((\alpha,\beta)=(0.5,0.5)\). Figure 7: Histograms of Gibbs samples for parameters and censored lifetimes when \(n=50\) and \((\alpha,\beta)=(0.5,0.5)\). (3) When it comes to the results of MLE of each parameter, the performance of the two methods, namely the optimization method and the EM algorithm, is quite similar. (4) In the situation where the sample size \(n\) is limited (e.g., \(n=30\)), Bayesian estimators exhibit lower MSEs compared to MLEs, which indicates that, in small-sample scenarios, the Bayesian method outperforms the classical frequentist method. (5) As expected, the Bayesian estimators under informative prior distribution perform better than those under non-informative prior distribution. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & \multicolumn{2}{c}{**MLE**} & \multicolumn{2}{c}{**BE**} \\ \cline{4-8} \multicolumn{1}{c}{(\(\alpha\), \(\beta\))} & \multicolumn{1}{c}{**pc**} & \multicolumn{1}{c}{\(n\)} & \multicolumn{1}{c}{**OPT**} & \multicolumn{1}{c}{**EM**} & \multicolumn{1}{c}{**Prior 1**} & \multicolumn{1}{c}{**Prior 2**} \\ \hline (0.5, 0.5) & 0.19 & 30 & bias & 0.0241 & 0.0241 & \(-\)0.0503 & 0.0099 \\  & & & mse & 0.0328 & 0.0330 & 0.0348 & 0.0282 \\  & & 50 & bias & 0.0101 & 0.0100 & \(-\)0.0510 & 0.0010 \\  & & & mse & 0.0166 & 0.0166 & 0.0310 & 0.0174 \\  & & 100 & bias & 0.0078 & 0.0077 & \(-\)0.0524 & \(-\)0.0049 \\  & & & mse & 0.0077 & 0.0077 & 0.0265 & 0.0071 \\ \hline (0.5, 1) & 0.14 & 30 & bias & 0.0188 & 0.0186 & \(-\)0.0421 & 0.0019 \\  & & & mse & 0.0309 & 0.0311 & 0.0362 & 0.0246 \\  & & 50 & bias & 0.0102 & 0.0101 & \(-\)0.0501 & 0.0016 \\  & & & mse & 0.0165 & 0.0165 & 0.0332 & 0.0154 \\  & & 100 & bias & 0.0074 & 0.0073 & \(-\)0.0640 & \(-\)0.0011 \\  & & & mse & 0.0077 & 0.0077 & 0.0361 & 0.0076 \\ \hline (1, 0.5) & 0.28 & 30 & bias & 0.0195 & 0.0195 & \(-\)0.0553 & \(-\)0.0036 \\  & & & mse & 0.0318 & 0.0318 & 0.0399 & 0.0253 \\  & & 50 & bias & 0.0104 & 0.0103 & \(-\)0.0509 & \(-\)0.0098 \\  & & & mse & 0.0169 & 0.0169 & 0.0317 & 0.0148 \\  & & 100 & bias & 0.0081 & 0.0080 & \(-\)0.0784 & \(-\)0.0111 \\  & & & mse & 0.0078 & 0.0078 & 0.0382 & 0.0076 \\ \hline \hline \end{tabular}
\end{table}