Entropy is a scientific term that finds applications in various domains, such as the laws of thermodynamics, where it was initially discovered, as well as statistical physics and information theory. We used unified hybrid censored data to investigate some inverse Weibull distribution entropy metrics. Entropy is defined using three measures: Rényi, Shannon, and Tsallis entropy. The classical estimates of the entropy measures were developed using the unified hybrid censored data, which included both point and approximation confidence intervals. The Bayesian method utilized the Markov Chain Monte Carlo sampling technique to develop Bayesian estimations. This was done by employing two loss functions, namely squared error and general entropy loss functions. Additionally, we delved into the investigation of Bayes credible intervals. Monte Carlo simulations were applied to explain how the estimates functioned at different sample sizes and censoring strategies via some accuracy criteria. Several observations were made in light of the simulation results. To provide a clear explanation of the offered methodologies, two applications using mechanical and cancer data sets were investigated.

The Gompertz–Makeham (GM) distribution has the flexibility to model real-world lifetime data with increasing, decreasing, or constant hazard rates, making it exceptionally valuable for applications in survival analysis, actuarial science, demography, and reliability engineering. This study proposes and rigorously analyzes a novel discrete formulation of the classical GM distribution, tailored to address real-world applications where event times are inherently discrete. Utilizing the survival function discretization technique, the authors derive the discrete GM (DGM) model and establish its foundational probability mass function, hazard rate function, and cumulative distribution function. A comprehensive suite of statistical properties—including quantiles, moments, skewness, kurtosis, and order statistics—is developed and examined numerically. Recognizing the challenges of parameter estimation under Type–Ⅱ data censoring, the paper implements both maximum likelihood estimation and Bayesian inference, with the latter incorporating gamma priors and executed via a Metropolis–Hastings Markov chain Monte Carlo algorithm. The paper further evaluates the estimator's performance through extensive simulations. The findings consistently demonstrate the superiority of Bayesian methods, particularly with high posterior density intervals. From three life sciences, several empirical case studies underscore the practical utility of the DGM model, showcasing improved goodness-of-fit relative to existing discrete models, for example, the discrete Nadarajah–Haghighi, discrete modified Weibull, discrete Weibull, and discrete gamma models, among others. Finally, this work fills a notable gap in the literature by extending the GM framework to discrete domains with full inferential machinery.

Modern products often have long life cycles and high reliability, making it difficult to collect comprehensive product life data with all unit failures for reliability and quality analysis. So, a new sampling plan called the generalized Type-Ⅱ progressive hybrid censored strategy has been suggested to minimize test time and costs. This study introduces a novel statistical framework for modeling lifetime data under generalized progressive hybrid censoring using the log-logistic (LogL) lifespan model. Besides traditional methodologies, our approach integrates frequentist and Bayesian inferential techniques to estimate key parameters and reliability metrics, such as the survival and hazard functions of the LogL distribution. The relevant approximate confidence intervals for unknown numbers are also constructed using the frequentest estimators' normal approximations. Incorporating the Markovian technique into Bayesian analysis, we leverage independent gamma priors and the Metropolis-Hastings algorithm to enhance computational efficiency to calculate the Bayes' point estimators along with their highest posterior density interval estimators. Additionally, we propose an optimal progressive censoring scheme that minimizes experimental costs while maintaining estimation accuracy. Extensive Monte Carlo simulations confirm the superiority of the proposed estimators, while three real-world applications in physics and engineering demonstrate their practical efficacy. The findings highlight the versatility of the LogL model and its potential as a robust survival analysis tool under complex real-world conditions.

Discrete lifetime data arising in engineering reliability, biomedical studies, and risk-based applications often display nonmonotone aging behavior, heavy tails, and substantial overdispersion, which challenge the adequacy of classical discrete models. To overcome these limitations, the paper proposes a new discrete Pham (DPham) distribution obtained through survival-function discretization of the continuous Pham model. This construction yields a parsimonious yet highly flexible two-parameter discrete lifetime family that substantially extends the modeling capacity of existing discrete distributions. A central contribution of the DPham model is its ability to accommodate a wide range of hazard rate shapes, including decreasing, increasing, constant, and bathtub forms, within a single, unified framework. This flexibility allows the model to capture complex aging mechanisms such as early-failure dominance, intermediate-life stabilization, and long-tailed persistence while retaining analytical tractability. The probability mass function supports both monotone and unimodal configurations, enhancing its descriptive power in discrete settings. The paper develops comprehensive distributional properties of the DPham distribution, including quantile representation, dispersion characteristics, skewness, kurtosis, and order statistics, demonstrating a level of adaptability that exceeds that of commonly used discrete lifetime models. Inferential procedures are systematically explored under both likelihood-based and Bayesian frameworks, with explicit consideration of Type-Ⅱ censored data. Bayesian estimation, implemented using Markov iterative algorithms, is shown to provide improved estimation accuracy and more reliable interval estimates, particularly in small-sample and heavily censored scenarios. The practical utility of the DPham distribution is illustrated through applications to real datasets from clinical survival analysis, industrial reliability systems, and overdispersed risk data. Across all cases, the proposed model consistently outperforms nine established discrete competitors, including discrete Weibull-type and gamma-based models, in terms of goodness of fit, inferential stability, and robustness to extreme observations. Supported by extensive simulation evidence, these results establish the DPham distribution as a versatile and domain-agnostic tool for discrete lifetime modeling, offering a compelling alternative for reliability theory, biostatistics, and applied risk analysis.

This study developed several inferential procedures for a modified and highly flexible extension of the classical Weibull distribution, termed the very flexible Weibull (VFW) distribution. Statistical inference was conducted under an adaptive progressive censoring scheme, which enhances dynamic control over the test duration and sample utilization. Both likelihood-based and Bayesian estimation methods were established for the model parameters and associated reliability measures. Nonlinear likelihood equations were derived and solved numerically using the Newton–Raphson iterative method, and asymptotic confidence intervals were constructed under normal and log-normal approximations. Within the Bayesian framework, independent gamma priors were assumed, and posterior inference was carried out using Markov chain Monte Carlo simulation based on the Metropolis–Hastings algorithm to obtain posterior summaries and two types of credible intervals. A numerical assessment was conducted to evaluate the statistical performance of the proposed estimators under various adaptive censoring configurations and prior specifications. The simulation results demonstrated that Bayesian estimators, particularly under informative priors, provide superior bias reduction, improved coverage probabilities, and more stable interval estimates than their frequentist counterparts. To illustrate the practical applicability of the proposed methodology, two real engineering datasets were analyzed. The data analysis confirmed the excellent fitting capability of the VFW distribution for modeling complex lifetime behaviors. From an engineering standpoint, the data-driven outcomes underscore the VFW distribution as a flexible and dependable lifetime model that facilitates precise reliability evaluation and decision-making in actual industrial systems.

A novel inverted generalized gamma (IGG) distribution, proposed for data modelling with an upside-down bathtub hazard rate, is considered. In many real-world practical situations, when a researcher wants to conduct a comparative study of the life testing of items based on cost and duration of testing, censoring strategies are frequently used. From this point of view, in the presence of censored data compiled from the most well-known progressively Type-Ⅱ censoring technique, this study examines different parameters of the IGG distribution. From a classical point of view, the likelihood and product of spacing estimation methods are considered. Observed Fisher information and the delta method are used to obtain the approximate confidence intervals for any unknown parametric function of the suggested model. In the Bayesian paradigm, the same traditional inferential approaches are used to estimate all unknown subjects. Markov-Chain with Monte-Carlo steps are considered to approximate all Bayes’ findings. Extensive numerical comparisons are presented to examine the performance of the proposed methodologies using various criteria of accuracy. Further, using several optimality criteria, the optimum progressive censoring design is suggested. To highlight how the proposed estimators can be used in practice and to verify the flexibility of the proposed model, we analyze the failure times of twenty mechanical components of a diesel engine.