## 1 Introduction  
A system's reliability function, \(R(t)\), represents the probability that an object survives longer than time \(t\), which is also called the survival function. Reliability analyses are mainly devoted to forecasting the probability of survival time for experimental systems, which can help administrators control costs, plan maintenance actions, and assess system availability. Based on the needs of practical applications, numerous scholars have investigated the reliability of different distributions, making use of classical estimators and Bayesian estimators under various loss functions and different priors. The authors of [1] provided the maximum-likelihood and Bayesian estimates for the three-parameter exponential-Weibull distribution with type-II censored samples. The authors of [2] analyzed the parameter estimates and reliability features of the one-parameter Lindley distribution using classical and Bayesian techniques with progressive type-II censored samples. A novel parameter estimation method named the E-Bayesian method was introduced in [3], which was used to estimate the reliability under a binomial distribution. Under a hybrid model with two exponential distributions of certain mixing proportions, the authors of [4] investigated the unbiased estimations of the reliability function. Furthermore, the equivalent unbiased estimators of the reliability were also derived under the scenario where the negative weighting of the components of the mixture distribution was allowed. The authors of [5] considered an estimator of the reliability function of the power Lomax distribution in stress-strength models. The authors of [6] obtained the classical reliability estimations of a multi-component system, in which the stress and strength variables followed the unit-generalized exponential distributions. The Bayesian estimations with Gamma and weighted Lindley priors were also discussed. On account of the situation where the data of torpedo loading tests were limited and included samples without failures, the authors of [7] proposed a new Bayesian evaluation method based on fused information to estimate the reliability of torpedo loading, for which the lifetime distribution and failure rate prior distribution were assumed to follow exponential and Gamma distributions, respectively. The authors of [8] calculated the estimators of the reliability for the adaptive, type-II progressive censored data of the unit-Lindley distribution using frequentist and Bayesian methods. In contrast to the limitations of classical estimation methods, which exclude prior information, Bayesian estimates have been more abundantly researched by scholars in recent years due to their consideration of prior distributions, allowing for the inclusion of historical or subjective information. In Bayesian methods, estimates are obtained using different loss functions. Therefore, one of the main challenges for researchers is choosing the right loss functions according to the specific situation; that is, a well-designed loss function yields a more accurate Bayesian estimator. Many scholars have investigated new loss functions and compared the performance of Bayesian estimators under different loss functions. The authors of [9] discussed the Bayesian parameter estimator of the Lindley distribution and their corresponding posterior risks separately using seven different loss functions. However, in response to the derivation error of the posterior risks under the entropy loss function in the paper, a correct derivation was given in [10]. The authors of [11] proposed a novel analysis method to evaluate potential loss in a process safety field. The potential loss analysis was discussed in detail along with case studies. The authors of [12] suggested defining a new loss function for use in Bayesian methods to decrease the effects of outliers in crowd behavior analysis.