## 5 Simulation and Data Analysis
### Real Data Analysis  
We apply real-world data from [9] in this subsection to demonstrate the inference framework developed in this study. The real dataset is from the National Eye Institute diabetic retinopathy study, whose purpose is to examine the influence of laser treatment on the delay of the start of blindness in those suffering from diabetic retinopathy. With respect to the \(i\)-th participant, \(X_{i,1}\) represents the blind time of the eye which has received laser treatment, while \(X_{i,2}\) refers to the blind time of the untreated eye. Then, the time of first onset of blindness \(T_{i}=\min\{X_{i,1},X_{i,2}\}\). Therefore, this dataset exhibits characteristics of competing risks data. Original data with corresponding cause are in Table 8. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{(\alpha,\beta)}\)} & \multirow{2}{*}{**pc**} & \multirow{2}{*}{\(n\)} & \multicolumn{2}{c}{**ACI**} & \multicolumn{2}{c}{**HPD-P1**} & \multicolumn{2}{c}{**HPD-P2**} \\ \cline{5-10}  & & & **AW** & **CP** & **AW** & **CP** & **AW** & **CP** \\ \hline (0.5, 0.5) & 0.19 & 30 & 0.9264 & 0.9380 & 0.7919 & 0.8640 & 0.8571 & 0.9390 \\  & & 50 & 0.7095 & 0.9410 & 0.6088 & 0.8520 & 0.6638 & 0.9350 \\  & & 100 & 0.4987 & 0.9500 & 0.4309 & 0.8550 & 0.4719 & 0.9200 \\ \hline (0.5, 1) & 0.14 & 30 & 0.9191 & 0.9370 & 0.7939 & 0.8800 & 0.8519 & 0.9320 \\  & & 50 & 0.7044 & 0.9420 & 0.6082 & 0.8730 & 0.6656 & 0.9300 \\  & & 100 & 0.4951 & 0.9530 & 0.4166 & 0.8210 & 0.4729 & 0.9400 \\ \hline (1, 0.5) & 0.28 & 30 & 0.9374 & 0.9460 & 0.7697 & 0.8500 & 0.8484 & 0.9100 \\  & & 50 & 0.7170 & 0.9400 & 0.5983 & 0.8360 & 0.6563 & 0.9320 \\  & & 100 & 0.5035 & 0.9580 & 0.4046 & 0.8020 & 0.4671 & 0.9210 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Average width and coverage percentage for interval estimation of \(a_{2}\) when \(\gamma=0.05\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{(\alpha,\beta)}\)} & \multirow{2}{*}{**pc**} & \multirow{2}{*}{\(n\)} & \multicolumn{2}{c}{**ACI**} & \multicolumn{2}{c}{**HPD-P1**} & \multicolumn{2}{c}{**HPD-P2**} \\ \cline{5-10}  & & & **AW** & **CP** & **AW** & **CP** & **AW** & **CP** \\ \hline (0.5, 0.5) & 0.19 & 30 & 0.6830 & 0.9550 & 0.5951 & 0.9050 & 0.6172 & 0.9410 \\  & & 50 & 0.5181 & 0.9610 & 0.4467 & 0.8630 & 0.4705 & 0.9240 \\  & & 100 & 0.3592 & 0.9510 & 0.3064 & 0.8600 & 0.3296 & 0.9410 \\ \hline (0.5, 1) & 0.14 & 30 & 0.6621 & 0.9430 & 0.5839 & 0.8920 & 0.6111 & 0.9430 \\  & & 50 & 0.5043 & 0.9550 & 0.4386 & 0.8740 & 0.4685 & 0.9260 \\  & & 100 & 0.3497 & 0.9660 & 0.2916 & 0.8240 & 0.3269 & 0.9460 \\ \hline (1, 0.5) & 0.28 & 30 & 0.7239 & 0.9760 & 0.6167 & 0.8630 & 0.6341 & 0.9140 \\  & & 50 & 0.5486 & 0.9820 & 0.4585 & 0.8450 & 0.4906 & 0.9040 \\  & & 100 & 0.3783 & 0.9790 & 0.3003 & 0.7860 & 0.3416 & 0.8880 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Average width and coverage percentage for interval estimation of \(b\) when \(\gamma=0.05\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Unit: Days** & & & & & & & \\ \hline (266, 1) & (583, 1) & (79, 1) & (93, 1) & (805, 1) & (344, 1) & (306, 1) & (415, 1) \\ (178, 1) & (1484, 1) & (315, 1) & (1252, 1) & (642, 1) & (407, 1) & (356, 1) & (699, 1) \\ (667, 1) & (126, 1) & (350, 1) & (84, 1) & (392, 1) & (901, 1) & (276, 1) & (520, 1) \\ (503, 1) & (584, 1) & (355, 1) & (1302, 1) & (91, 2) & (154, 2) & (547, 2) & (707, 2) \\ (469, 2) & (1313, 2) & (790, 2) & (125, 2) & (777, 2) & (307, 2) & (637, 2) & (577, 2) \\ (517, 2) & (287, 2) & (717, 2) & (141, 2) & (427, 2) & (36, 2) & (588, 2) & (350, 2) \\ (567, 2) & (1140, 2) & (448, 2) & (904, 2) & (485, 2) & (248, 2) & (423, 2) & (285, 2) \\ (315, 2) & (727, 2) & (210, 2) & (409, 2) & (227, 2) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The original data of the blind time with corresponding cause. To ensure that the solution exists, we divide the data by 365 to let the unit be year.