## 5 Simulation and Data Analysis
### Real Data Analysis
It should be noted that, before further analysis, we need to first determine whether the dataset is able to be analyzed using the Burr-XII distribution. Separately for failure cause 1 and 2, initially, the complete dataset is employed to derive MLEs of unknown parameters, followed by the application of the Kolmogorov-Smirnov test. For cause 1, the MLEs of \(a_{1}\) and \(b_{1}\) are 0.9033 and 2.3703, respectively. The value of Kolmogorov-Smirnov statistics is 0.1168 with its corresponding \(p\)-value \(=0.7978\). For cause 2, the MLEs of \(a_{2}\) and \(b_{2}\) are 0.9237 and 2.4605, respectively. The value of Kolmogorov-Smirnov statistics equals 0.1470 with its corresponding \(p\)-value amounting to 0.4324. Consequently, we can conclude that the Burr-XII distribution can be used to fit the dataset mentioned above. In addition, \(b_{1}=b_{2}=b\) need to be tested. We use the likelihood ratio test method. Through the analysis and calculation of the sample data, the corresponding \(p\)-value is quite close to 1, which indicates that the hypothesis \"\(b_{1}=b_{2}=b\)\" is reasonable. Then, we generate the artificial data. Both the left endpoint and the interval length are the random variables which follow the exponential distribution with mean 1. The processed data (measured in years) are presented in Table 9.  
Since we have no extra information of \(a_{1}\), \(a_{2}\), and \(b\), we employ the non-informative priors for them. Using the methods and procedures of estimation outlined above, point and interval estimates for all unknown parameters are derived, with results shown in Table 10, where '\(MLE_{OPT}\)', '\(MLE_{EM}\)' represent the MLEs obtained by the \(optim\) function or the EM algorithm, respectively, and 'BE' refers to the Bayesian point estimator. We can see that the results from two methods to obtain MLEs are same and the widths of HPDs are less than those of ACIs. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(MLE_{OPT}\) & \(MLE_{EM}\) & BE & **ACI** & **HPD** \\ \hline \(a_{1}\) & 0.4339 & 0.4339 & 0.3025 & [0.2584, 0.6095] & [0.1832, 0.4357] \\ \(a_{2}\) & 0.5114 & 0.5114 & 0.3571 & [0.3182, 0.7047] & [0.1664, 0.4951] \\ \(b\) & 2.3292 & 2.3292 & 2.0231 & [1.6961, 2.9624] & [1.5444, 2.5151] \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The result of different point and interval estimates for \(a_{1},a_{2},b\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Cause** & \multicolumn{4}{c}{**Data (Years)**} \\ \hline Cause 1 & 0.7288 & 0.2164 & 0.2548 & 2.2055 \\  & 0.9425 & 1.1370 & 4.0658 & 1.0740 \\  & 0.8630 & 3.4301 & 1.7589 & 1.1151 \\  & 1.9151 & 1.8274 & 0.9589 & 0.2301 \\  & 0.9726 & 3.5671 & 0.7562 & 1.4247 \\  & 1.6000 & [0.9011, 5.0378] & [0.6011, 1.7187] & [0.3192, 0.6168] \\  & [0.8229, 1.0968] & [0.0267, 0.5380] & [2.0658, 4.4214] & [0.1413, 2.1862] \\ \hline Cause 2 & 0.2493 & 1.4986 & 1.9370 & 1.2849 \\  & 3.5973 & 2.1644 & 0.3425 & 2.1288 \\  & 1.7452 & 1.5808 & 1.9644 & 0.3863 \\  & 1.1699 & 0.0986 & 1.5534 & 1.2274 \\  & 2.4767 & 1.1589 & 0.7808 & 0.8630 \\  & 1.9918 & 0.5753 & 1.1205 & 0.6219 \\  & [0.1226, 0.9680] & [0.6727, 2.0312] & [0.4918, 2.2852] & [0.4439, 0.9947] \\  & [0.7571, 2.0435] & [0.0610, 1.6743] & [0.9411, 3.4598] & [0.3921, 1.3314] \\  & [0.1619, 1.0728] & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The artificial middle-censored data. Based on the estimated values of parameters \(a_{1},a_{2}\), and \(b\), we can draw plots that combine the fitted probability density curves with the frequency histograms of the complete data for the two causes, respectively. In Figures 9 and 10, 'MLE' denotes the density curve based on MLEs under middle-censored data, 'BE' denotes the curve based on Bayes estimates under middle-censored data, and 'MLE_c' represents the density curve with parameter estimates derived from the complete data.