###### Abstract  
The generalized Pareto distribution plays a significant role in reliability research. This study concentrates on the statistical inference of the generalized Pareto distribution utilizing progressively Type-II censored data. Estimations are performed using maximum likelihood estimation through the expectation-maximization approach. Confidence intervals are derived using the asymptotic confidence intervals. Bayesian estimations are conducted using the Tierney and Kadane method alongside the Metropolis-Hastings algorithm, and the highest posterior density credible interval estimation is accomplished. Furthermore, Bayesian predictive intervals and future sample estimations are explored. To illustrate these inference techniques, a simulation and practical example are presented for analysis. generalized Pareto distribution; expectation-maximization algorithm; progressive Type-II censoring; Metropolis-Hasting approach; Bayesian estimation; Bayesian prediction 2024 14 8433 10.3390/00000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-00000-0000-00000-0000-00000-00000-00000-00000-00  
The Generalized Pareto Distribution (GPD) is extensively used in life test analysis across multiple domains, including biology and geography. For instance, the study by [5] explored the applications of GPD in wind engineering, highlighting its advantages over the generalized extreme value distribution in the context of extreme value analysis. Additionally, ref. [6] investigated the effects of event independence and threshold selection by conducting a GPD analysis on gust velocity maxima recorded at several island locations. In this paper, we utilize the generalized Pareto distribution with unknown parameters, as discussed by [7], to estimate these parameters using progressive Type-II censoring samples and to make corresponding Bayesian predictions. In recent years, numerous researchers have made significant contributions to the study of the parameters of the generalized Pareto distribution. For instance, ref. [8] investigated minimum density power divergence estimations (MDPDE) for GPD parameters and conducted a comparison of the efficiencies among MDPDE, maximum likelihood estimations (MLE), [PERSON]'s optimally biased robust estimator, and the medians estimator proposed by [PERSON] and [PERSON]. Additionally, ref. [9] addressed statistical inference challenges related to GPD within the framework of progressive Type-I censoring. The author successfully derived MLEs using the Expectation-Maximization (EM) algorithm along with the Fisher information matrix, and they also formulated asymptotic confidence intervals for the parameters. The discussion in [10] focused on estimating GPD parameters through MLE, the probability-weighted moments method, and the method of moments. Moreover, ref. [11] implemented bootstrap algorithms and Monte Carlo methods to derive Bayesian estimations and confidence intervals for GPD. Furthermore, ref. [12] utilized LINEX, entropy loss, and precautionary functions to achieve Bayesian estimations of GPD parameters, employing quasi, uniform prior, and inverted gamma distributions. For further details and examples related to the generalized Pareto distribution, please refer to the following articles: [6; 7; 9; 13; 14; 15; 16]. The remainder of this article is structured as follows: Section 2 will discuss Maximum Likelihood Estimators (MLEs). Section 3 outlines the observed Fisher information matrix and the methodology for calculating asymptotic confidence intervals. In Section 4, we derive Bayesian estimations of unknown parameters utilizing various loss functions. These estimations are then computed using the Tierney and Kadane (TK) method. Section 4 also introduces the Metropolis-Hastings (MH) algorithm, which is employed to obtain Bayesian estimations. The samples generated using this approach are utilized to construct the highest posterior density intervals. In Section 5, we present Bayesian point predictions and interval predictions for future observable samples.