## 7 Conclusions  
Originating from the QEL function, we have proposed a new loss function called the WQEL and discuss the corresponding parameter and reliability function estimators of the Burr XII distribution under it. Classical estimators, including the maximum-likelihood and uniformly minimum-variance unbiased estimations, were also derived comprehensively. Considering Gamma and quasi-density priors, we discussed the estimators under the SEL, GEL, and QEL functions for Bayesian procedures and compared them with the estimators under the new loss function. Through Monte Carlo simulation, the estimator under the WQEL function was found to be superior to the other estimators, as can be seen in the tables and plots depicting the discussed estimators of the reliability function. Then, a remission time data set was analyzed to verify the applicability of the estimators introduced in this study. Before handling the data set with different estimators, we carried out BIC, Kolmogorov-Smirnov testing, and histogram procedures to confirm that the distribution underlying the real data was a Burr XII distribution. Finally, we presented the estimators of the reliability for the adopted data set. We investigated frequentist and Bayesian inferences of the reliability function under the Burr XII distribution with a single unknown parameter and complete samples in this study. In future research, we expect to explore the corresponding estimators of the Burr XII distribution with double unknown parameters, as well as various censored schemes. We will also investigate the performance and provide specific guidance regarding the parameters of the new loss function in more cases and models. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(t=0.5\) & \(t=1\) & \(t=1.5\) & \(t=2\) & \(t=3\) & \(t=4\) & **Running Time (s)** \\ \hline \(\hat{R}_{MLE}(t)\) & 0.520860 & 0.239044 & 0.120704 & 0.067438 & 0.026501 & 0.012836 & 0.192488 \\ \(\hat{R}_{LUMIUE}(t)\) & 0.523521 & 0.241732 & 0.122715 & 0.068874 & 0.027264 & 0.013280 & 0.185871 \\ \(\hat{R}_{SEL}(t)\) & 0.520399 & 0.239615 & 0.121806 & 0.068634 & 0.027488 & 0.013572 & 0.189641 \\ \(\hat{R}_{GEL}(t)\) & 0.517792 & 0.233867 & 0.115495 & 0.062928 & 0.023466 & 0.010798 & 0.187371 \\ \(\hat{R}_{QEL}(t)\) & 0.519522 & 0.237655 & 0.119619 & 0.066624 & 0.026026 & 0.012534 & 0.198095 \\ \(\hat{R}_{WQEL}(t)\) & 0.521250 & 0.241462 & 0.123822 & 0.070458 & 0.028787 & 0.014483 & 0.212756 \\ \hline \end{tabular}
\end{table} 
Table 7: The estimates of the reliability for selected points of the data set along with the corresponding running times. * [1] [PERSON]; [PERSON]; [PERSON] Estimation of three-parameter exponentiated-Weibull distribution under type-II censoring. _J. Stat. Plan. Inference_**2005**, _134_, 350-372. [CrossRef]
* [2] [PERSON]; [PERSON] Reliability estimation in Lindley distribution with progressively type II right censored sample. _Math. Comput. Simul._**2011**, _82_, 281-294. [CrossRef]
* [3] [PERSON] E-Bayesian estimation of the reliability derived from Binomial distribution. _Appl. Math. Model._**2011**, _35_, 2419-2424. [CrossRef]
* [4] [PERSON]; [PERSON]; [PERSON] Unbiased estimation of reliability function from a mixture of two exponential distributions based on a single observation. _Stat. Probab. Lett._**2017**, _127_, 7-13. [CrossRef]
* [5] [PERSON]; [PERSON] On estimation of the stress-strength reliability on POLO distribution function. _Ain Shams Eng. J._**2021**, _12_, 4037-4044. [CrossRef]
* [6] [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Multicomponent Stress-Strength Reliability estimation based on Unit Generalized Exponential Distribution. _Ain Shams Eng. J._**2022**, _13_, 101627. [CrossRef]
* [7] [PERSON]; [PERSON]; [PERSON]; [PERSON] Reliability evaluation method of torpedo loading based on zero-failure data. _Ocean. Eng._**2023**, _278_, 114431. [CrossRef]
* [8] [PERSON]; [PERSON]; [PERSON]; [PERSON] Optimal analysis of adaptive type-II progressive censored for new unit-lindey model. _J. King Saud Univ. Sci._**2023**, _35_, 102462.