In this paper the exponentiated generalized xgamma distribution is introduced. Some of its properties are presented through some models of stress–strength, moments, mean residual life, mean past lifetime, and order statistics. The maximum likelihood estimators, confidence intervals for the parameters, and the reliability and the hazard rate functions of the exponentiated generalized xgamma distribution based on dual generalized order statistics are obtained. Bayesian estimators for the unknown parameters, reliability, and hazard rate functions of the exponentiated generalized xgamma distribution based on dual generalized order statistics are considered. The results based on lower record values are verified using simulations as well as three real sets of data are adopted to demonstrate the flexibility and potential applications of the distribution.

In lifetime tests, the waiting time for items to fail may be long under usual use conditions, particularly when the products have high reliability. To reduce the cost of testing without sacrificing the quality of the data obtained, the products are exposed to higher stress levels than normal, which quickly causes early failures. Therefore, accelerated life testing is essential since it saves costs and time. This paper considers constant stress-partially accelerated life tests under progressive Type II censored samples. This is realized under the claim that the lifetime of products under usual use conditions follows Vtub-shaped lifetime distribution, which is also known as log-log distribution. The log–log distribution is highly significant and has several real-world applications since it has distinct shapes of its probability density function and hazard rate function. A graphical description of the log–log distribution is exhibited, including plots of the probability density function and hazard rate. The log–log density has different shapes, such as decreasing, unimodal, and approximately symmetric. Several mathematical properties, such as quantiles, probability weighted moments, incomplete moments, moments of residual life, and reversed residual life functions, and entropy of the log–log distribution, are discussed. In addition, the maximum likelihood and maximum product spacing methods are used to obtain the interval and point estimators of the acceleration factor, as well as the model parameters. A simulation study is employed to assess the implementation of the estimation approaches under censoring schemes and different sample sizes. Finally, to demonstrate the viability of the various approaches, two real data sets are investigated.

Recent advances in lifetime modeling have led to several extensions of classical distributions, among which the extended inverted Kumaraswamy lifetime model, known as the exponentiated generalized inverted Kumaraswamy model, represents a significant innovation. In parallel, progressive Type-II censoring frameworks have garnered growing attention for their adaptability and relevance across various applied disciplines, including medical, engineering, and social sciences. Driven by this motivation, the present study focuses on the problem of parameter estimation for the proposed lifetime model under a progressive Type-II censoring scheme. Both Bayesian and non-Bayesian frameworks are utilized to estimate the model parameters, reliability function, and hazard rate. Furthermore, interval estimation is conducted by constructing confidence and Bayesian credible intervals for these measures. Assuming independent gamma priors, Bayes estimators are derived under both symmetric and asymmetric loss functions to account for different decision-making perspectives. In addition, conditional and Bayesian predictive analyses are developed within a two-sample prediction framework, and their associated prediction intervals are constructed. The efficiency and robustness of the proposed estimation and prediction methodologies are thoroughly evaluated through an extensive simulation study conducted under various sample sizes and censoring schemes. To further demonstrate the model’s practical relevance, real-world medical and engineering datasets are analyzed, highlighting the applicability and effectiveness of the proposed distribution in empirical contexts.

Entropy is a fundamental measure of uncertainty in reliability and lifetime analysis, and the structure of the observed data intrinsically influences its estimation. In many practical applications, inference relies on ordered or record-based samples, for which generalized order statistics and their dual form provide a comprehensive and unifying framework encompassing order statistics, reversed order statistics, and record values as special cases. Despite substantial progress in entropy estimation and lifetime modeling, little attention has been devoted to a unified entropy estimation framework for flexible lifetime models under generalized ordered sampling schemes. This paper investigates the impact of generalized order statistics and their dual form on the estimation of entropy measures for the exponentiated generalized Pham distribution. The proposed distribution extends the classical Pham model through additional shape flexibility, enabling it to accommodate diverse reliability behaviors and heterogeneous tail characteristics. Several fundamental properties are derived, and maximum likelihood estimation and corresponding confidence intervals for model parameters and entropy measures are developed under both generalized order statistics and dual generalized order statistics frameworks. The general results are further specialized to order statistics, reversed order statistics, and upper and lower record values. Applications to two real datasets demonstrate that the proposed distribution provides an excellent fit compared with competing models, as confirmed by goodness-of-fit measures. The findings underscore the pivotal structural role of generalized order statistics and their dual form in entropy-based inference, particularly for record or partially observed data, where the sampling design critically affects uncertainty quantification.