## 1 Introduction
The authors of [13] proposed a novel loss function named the weighted composite Linex loss function and compared the performance of the estimators under it with other estimators under the Linex loss, weighted Linex loss, and composite Linex loss functions. The authors of [14] introduced a novel weighted general entropy loss function and compared it with other loss functions. The authors of [15] derived the parameter estimations of the exponential distribution with complete data, which were comparatively analyzed using maximum-likelihood and Bayesian estimations with the squared error loss, entropy loss, and composite Linex loss functions. The Burr XII distribution is very flexible as a lifetime distribution, as its hazard rate function can be either monotonic or unimodal, which means that it is much more suitable for modeling different types of lifetime data. Thus, it is more compatible with the precise situations of system reliability analysis and has been widely studied by scholars and applied in practice. The probability density function (PDF) and cumulative distribution functions are expressed as  
\[f(x;\alpha,\beta)=\alpha\beta x^{\alpha-1}(1+x^{\alpha})^{-\beta-1},x>0,\alpha,\beta>0, \tag{1}\]  
\[F(x;\alpha,\beta)=1-(1+x^{\alpha})^{-\beta}, \tag{2}\]  
where \(\alpha\) and \(\beta\) represent the shape parameters. The hazard rate and reliability functions are expressed as  
\[h(x;\alpha,\beta)=\frac{\alpha\beta x^{\alpha-1}}{1+x^{\alpha}}, \tag{3}\]  
\[R(t;\alpha,\beta)=(1+t^{\alpha})^{-\beta}. \tag{4}\]  
As a vitally flexible distribution, the Burr XII distribution has many tight connections with other well-known distributions. When \(\alpha=1\), \(\beta=1\), and \(\alpha=\beta\), the Burr XII distribution is equivalent to the Pareto, log-logistic, and paradigistic distributions, respectively, with all their scale parameters equal to 1. The PDF plots corresponding to the selected parameter values are shown in Figure 1.  
Based on Figure 1, when \(\alpha>1\), the PDF plots of the Burr XII distribution have a single peak, which is skewed to the right. As \(\beta\) increases, the density plot becomes steeper. In contrast to the case of \(\alpha>1\), the function curve monotonically decreases when \(\alpha\leqslant 1\). The hazard rate plots with different values of \(\alpha\) are shown in Figure 2, indicating that the hazard rate decreases monotonically for \(\alpha\leqslant 1\). However, when \(\alpha>1\), the hazard rate quickly reaches its peak and then decreases as the argument \(x\) increases. As \(x\) approaches\(\infty\), the hazard rate value converges to a constant value. Moreover, the hazard rate has higher values when \(\beta\) is larger. The reliability plots are shown in Figure 3.  
The Burr XII distribution has a broad range of applications in lifetime analysis due to its excellent flexibility and practicality as a lifetime distribution. The Burr XII distribution was adopted in [16] to investigate the reliability of a transit traffic control system with medium loads. Using a progressive type-II censored scheme, the authors of [17] considered estimates for the Burr XII distribution using E-Bayesian methods. The authors of [18] discussed the Bayesian estimators of the parameters and reliability under the Burr XII distribution, which were compared with classical maximum-likelihood estimation. For a mixture system of three Burr XII distributions, the authors of [19] analyzed the Bayesian and maximum-likelihood estimators, along with the elegant closed-form algebraic expressions  
Figure 1: PDF plots of the Burr XII distribution for varying parameter values. Figure 3: Reliability plots of the Burr XII distribution with different values of \(\alpha\). Figure 2: Hazard rate plots of the Burr XII distribution with different values of \(\alpha\). of Bayesian estimators, as well as the posterior risks. Under the Burr XII distribution, the authors of [20] introduced an improved adaptive type-II progressive censoring scheme, in which the frequentist and Bayesian inferences were established.