## 5 Simulation
The null hypothesis for the test was that there was no significant difference between the two sets of data. The significance level was set at \(0.05\). Based on the results in Tables 2 and 5, the p-value of the Wilcoxon signed-rank test was calculated as \(p=0.7829>0.05\), indicating that the estimates under the SEL and WQEL function with \(q=0.5\), \(z=-0.5\) had similar values. Thus, when \(q=0.5\), \(z=-0.5\), the estimators under the WQEL function were equivalent to those under the SEL function. This demonstrates the flexibility and advantages of the new WQEL function. 3. As seen in Table 1, the classical estimators under the Burr XII distribution with parameter \(\alpha=2\) were more efficient than the estimators when \(\alpha=1\). 4. As seen in Tables 2-5, regardless of the loss function used, the estimates under the informative Gamma prior were more efficient than those under the non-informative quasi-density prior, which demonstrates the benefit of an appropriate informative prior distribution for Bayesian estimators. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & **Gamma Prior** & **Quasi-Density Prior (\(m=0.5\))** & **Quasi-Density Prior (\(m=1\))** \\ \(n\) & \(t\) & \(R(t)\) & \(\hat{R}_{SEL}(t)\) & \(\hat{R}_{SEL}(t)\) & \(\hat{R}_{SEL}(t)\) \\ \hline  & 0.5 & 0.89442719 & 0.87280694 & 0.87876107 & 0.88414680 \\  & & & (0.00005850) & (0.00281316) & (0.00212439) \\  & 1.0 & 0.70710678 & 0.66263949 & 0.67697393 & 0.68941180 \\  & 1.5 & 0.55470020 & 0.0044212 & (0.01368563) & (0.01039924) \\
10 & 2.0 & 0.44721360 & 0.50540811 & 0.52430887 & 0.54016925 \\  & & (0.00097336) & (0.01977326) & (0.01509506) \\  & 2.0 & 0.44721360 & 0.40146574 & 0.42221176 & 0.43920878 \\  & & (0.00140592) & (0.02012684) & (0.01540660) \\  & 3.0 & 0.31622777 & 0.28199122 & 0.30315901 & 0.31998567 \\  & 4.0 & 0.001689686 & (0.01583332) & (0.01214019) \\  & 4.0 & 0.24253563 & 0.21790019 & 0.23821225 & 0.25407353 \\  & & (0.00200895) & (0.01167751) & (0.00894554) \\ MISE & & 0.00112645 & 0.01398495 & 0.01068517 \\  & 0.5 & 0.89442719 & 0.88797084 & 0.89003636 & 0.89173440 \\  & & (0.00016410) & (0.00349479) & (0.00320910) \\  & 1.0 & 0.70710678 & 0.69349769 & 0.69843747 & 0.70253490 \\  & & (0.00089375) & (0.01778125) & (0.01640886) \\  & 1.5 & 0.55470020 & 0.53946625 & 0.54590884 & 0.55130046 \\  & 2.0 & 0.044721360 & (0.00143191) & (0.02675688) & (0.02480931) \\  & 2.0 & 0.44721360 & 0.43315695 & 0.44013600 & 0.44602126 \\  & & (0.00158133) & (0.02812639) & (0.02618147) \\  & 3.0 & 0.31622777 & 0.30620266 & 0.31313071 & 0.31904293 \\  & & (0.00139320) & (0.02316576) & (0.02168828) \\  & 4.0 & 0.24253563 & 0.23596620 & 0.24244581 & 0.24802471 \\  & 4.0 & 0.24253563 & 0.23596620 & 0.24244581 & 0.24802471 \\  & & (0.00110573) & (0.01763246) & (0.01657375) \\ MISE & & 0.00109500 & 0.01949292 & 0.01814513 \\  & 0.5 & 0.89442719 & 0.89185526 & 0.89300322 & 0.89350584 \\  & & (0.00012076) & (0.00032429) & (0.00033945) \\  & 1.0 & 0.70710678 & 0.70139532 & 0.70421845 & 0.70544587 \\  & & (0.00069190) & (0.00205874) & (0.00215655) \\  & 1.5 & 0.55470020 & 0.54789939 & 0.55167397 & 0.55330387 \\  & & (0.00116836) & (0.00386530) & (0.00405207) \\  & 2.0 & 0.44721360 & 0.44051295 & 0.44646979 & 0.44646880 \\  & & (0.00135234) & (0.00491583) & (0.00515697) \\  & 3.0 & 0.31622777 & 0.31070706 & 0.31494691 & 0.31675174 \\  & & (0.00128635) & (0.00544252) & (0.00571607) \\  & 4.0 & 0.24253563 & 0.23821968 & 0.24224326 & 0.24394571 \\  & & (0.00108394) & (0.00515261) & (0.00541648) \\  & MSE & & 0.00095061 & 0.00362655 & 0.00380627 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The Bayesian estimates under the SEL function and \(\alpha=2\) and \(\beta=0.5\), with their MSEs shown in brackets.