Osama E. Abo-Kasem

Modeling bounded data presents a fundamental challenge in reliability and risk analysis, particularly when the underlying observations exhibit heterogeneous distributional shapes and complex failure-rate patterns. To address this challenge, we introduce a novel Unit Hjorth (UHj) distribution, obtained through an exponential transformation of the classical Hjorth model, which transfers its flexible reliability structure to the unit interval while retaining analytical tractability. A comprehensive theoretical investigation of the proposed model is developed, including its boundary behavior, limiting submodels, quantile function, ordinary moments, cumulants, mode characterization, order statistics, stress–strength reliability, and other important reliability measures. The UHj density can be strictly increasing or non-monotone, including unimodal forms, while its hazard rate can accommodate increasing, bathtub-shaped, upside-down-bathtub, and modified-bathtub patterns. This broad hazard-rate flexibility makes the model particularly suitable for representing heterogeneous reliability and risk profiles that cannot be adequately captured by conventional bounded distributions. For statistical inference, a comprehensive estimation framework is established based on maximum likelihood, maximum product of spacings, and six additional classical estimation methods. Their finite-sample performance is systematically assessed through extensive Monte Carlo simulations using multiple measures of bias, efficiency, accuracy, and numerical stability. The simulation results indicate that the likelihood- and spacing-based procedures generally provide the most accurate and stable parameter estimates. The practical relevance of the proposed model is further demonstrated through three real-world datasets from computer science, engineering, and environmental applications. In these applications, the UHj distribution provides competitive, and in several cases superior, goodness-of-fit performance compared with a range of established unit distributions. Overall, the proposed model provides a unified and flexible framework for bounded-data modeling, reliability assessment, risk characterization, and statistical inference, offering a useful addition to the class of bounded probability models for complex applied-data scenarios.

The increasing availability of bounded lifetime observations in different disciplines has intensified the demand for flexible models capable of accommodating complex failure mechanisms. Motivated by this need, a comprehensive inferential framework is developed for the unit Bilal (UBilal) distribution using generalized progressive hybrid censoring, which guarantees a minimum number of observed failures while controlling experimental duration. Classical inference is established through maximum likelihood estimation, which is accompanied by asymptotic confidence intervals based on both normal and log-transformed approximations. Moreover, a Bayesian framework using a Metropolis–Hastings Markov chain Monte Carlo algorithm is presented. The proposed methodology further provides inference for important reliability characteristics, including the reliability and hazard rate functions, through both frequentist and Bayesian paradigms proposed. An extensive Monte Carlo investigation is conducted under diverse censoring schemes, sample sizes, and prior specifications to evaluate estimation accuracy, interval performance, and the influence of censoring severity. The simulation results show that Bayesian methods always provide better estimates and more reliable interval estimates, especially when prior information is used. Using two real datasets from thermodynamics and meteorology, the numerical results demonstrate that the UBilal model provides an excellent fit and yields reliable inference under bounded observations. Overall, the proposed methodology presents an efficient Bayesian inferential framework for bounded lifetime data collected through the generalized progressive hybrid censoring and expands the applicability of the UBilal model to reliability and related fields.

This article presents a comprehensive inferential and applied investigation of the newly reparameterized Z-Weibull (ZW) distribution, a flexible Weibull-type lifetime model capable of accommodating both bounded and unbounded support regimes as well as a wide variety of hazard rate shapes. Unified frequentist and Bayesian inference procedures are developed for complete and censored samples using maximum likelihood, maximum product spacing, and Markov chain Monte Carlo methods. Theoretical properties of the estimators and their associated interval estimates are established, while extensive Monte Carlo simulations assess their finite-sample performance under diverse parameter configurations and censoring schemes. The results indicate that Bayesian spacing-based procedures generally provide more accurate estimation, lower bias, and improved interval performance than competing classical methods. Applications to biomedical survival and climatological datasets, together with comparisons against several Weibull-type and exponential-based competitors, demonstrate the superior flexibility and goodness-of-fit of the ZW model. These findings highlight the practical value of the reparameterized ZW distribution as a unified and effective tool for modeling complex lifetime and reliability data arising in survival, environmental, and engineering studies.

This study presents a new version of the ZLindly (ZL) model that improves modeling flexibility while maintaining ease of analysis, allowing for the simultaneous accommodation of redundant zeros, thick-tailed behavior, and complex failure rate dynamics within a unified probabilistic framework. Marshall–Olkin (MO) theory facilitates this advancement. The MOZL hazard rate can exhibit several patterns, including increasing, decreasing, bathtub, or upside-down bathtub-shaped. These features enable the model to capture diverse reliability phenomena such as early-life failures, random shocks, and wear-out effects. Comprehensive theoretical investigations were conducted and shown to be governed by an interpretable dual-parameter mechanism, where the Marshall–Olkin parameter controls tail behavior and dispersion, while the scale parameter regulates skewness and hazard evolution. A likelihood-based approach was developed under Type-II censoring conditions, and rigorous evidence is provided for the existence and uniqueness. To address inferential uncertainty, both classical asymptotic confidence intervals and log-normal approximations were constructed. Within a Bayesian framework, independent gamma priors were assumed, and posterior inference was performed via an efficient Metropolis–Hastings algorithm. Bayesian point and credible estimators were obtained and compared with their classical counterparts. An extensive simulation study demonstrates that Bayesian estimators, particularly with informative priors, consistently outperform likelihood-based estimators in terms of bias, mean squared error, interval length, and coverage probability, especially for moderate sample sizes and higher censoring levels. Three engineering applications are provided to assess the practical utility of the MOZL model, where it provides superior goodness-of-fit relative to 15 competing models, including MO–Exponential, MO–Gompertz, MO–Nadarajah–Haghighi, MO–Exponentiated Weibull, and Birnbaum–Saunders, among others. Overall, the proposed MOZL distribution emerges as a flexible, interpretable, and computationally efficient lifetime model whose structurally meaningful parameter interactions enhance distributional balance and flexible hazard behavior, thereby contributing to modern symmetry-oriented distribution theory while offering valuable applications in reliability engineering, survival analysis, and applied statistical modeling.

In modern reliability engineering, modeling bounded lifetime data under realistic experimental conditions is still challenging, especially when censoring schemes and unit removals are random. This study proposes a new and unified reliability framework by combining the flexible powering new power (PNP) distribution with a grouping-based progressive first-failure mechanism using a beta-binomial random design. The proposed approach explicitly accounts for the randomness in group removals, providing a more realistic description of practical life-testing experiments. Classical estimation is carried out using maximum likelihood methods with the Newton-Raphson algorithm, along with confidence intervals constructed under both standard and log-transformed parameterizations. To increase flexibility in inference, a Bayesian approach is developed based on a joint gamma and shifted log-normal prior, which respects parameter constraints and incorporates prior uncertainty. Since the posterior distributions cannot be obtained in closed form, a Metropolis-Hastings Markov chain Monte Carlo algorithm is used to generate reliable posterior estimates and credible intervals. Additionally, beyond sensitivity analysis, multiple prior robustness diagnostics are incorporated to ensure reliable hyperparameter calibration and to safeguard against prior misspecification. The performance of the proposed estimators is carefully examined through extensive Monte Carlo simulations under different censoring schemes and parameter settings. The simulation results indicate that the proposed Bayesian procedures often provide more stable estimation and shorter interval estimates with competitive coverage probabilities compared with the corresponding classical methods, particularly under moderate-to-heavy censoring settings. To demonstrate its practical usefulness, the proposed model is applied to two real datasets on tensile strength of carbon and polyester fibers, where it provides a good fit and useful insights into material reliability and failure behavior. In the same applications, the practical relevance and superior performance of the proposed distribution are demonstrated, where it outperforms existing bounded versions of several well-known models, including the gamma, Weibull, and Birnbaum-Saunders distributions. Overall, this work contributes to reliability analysis by offering a flexible and computationally efficient framework that accounts for both random censoring and complex lifetime patterns, with potential applications in engineering, materials science, and applied reliability studies.

In modern reliability and survival analysis, modeling bounded-lifetime data under complex sampling mechanisms remains a challenging yet practically significant problem, particularly in biomedical and engineering applications where experimental time and cost constraints are critical. This study develops a comprehensive inferential framework for the unit-Lindley (ULind) distribution under the improved adaptive progressive Type-II censoring (I-AP-CT2) strategy, a flexible design that ensures bounded test duration while accommodating progressive removal of experimental units. The proposed approach integrates both classical and Bayesian paradigms to enable robust estimation of the model parameter, reliability function, and hazard rate function. Maximum likelihood estimators are derived, and their asymptotic properties are established, with interval estimation constructed via normal approximation and log-transformed techniques. To address analytical intractability, a Bayesian framework via MCMC-based is formulated with a gamma prior, yielding credible and highest posterior density intervals. An extensive Monte Carlo simulation study is conducted to evaluate estimator performance under diverse censoring scenarios, demonstrating that Bayesian procedures consistently outperform their frequentist counterparts in terms of accuracy, stability, and interval efficiency. The practical relevance of the proposed methodology is illustrated through two real-world applications involving kidney dialysis survival data and petroleum engineering reliability data, representing critical domains where accurate modeling of failure behavior directly impacts clinical decision-making and industrial risk management. The findings highlight the flexibility of the ULind model in capturing complex hazard rate shapes and confirm the effectiveness of the I-AP-CT2 mechanism as a realistic and efficient experimental design.

The Maxwell–Boltzmann (MB) distribution is important because it provides the statistical foundation for connecting microscopic particle motion to macroscopic gas properties by statistically describing molecular speeds and energies, making it essential for understanding and predicting the behavior of classical ideal gases. This study advances the statistical modeling of lifetime distributions by developing a comprehensive reliability analysis of the MB distribution under an improved adaptive progressive censoring framework. The proposed scheme strategically enhances experimental flexibility by dynamically adjusting censoring protocols, thereby preserving more information from test samples compared to conventional designs. Maximum likelihood estimation, interval estimation, and Bayesian inference are rigorously derived for the MB parameters, with asymptotic properties established to ensure methodological soundness. To address computational challenges, Markov chain Monte Carlo algorithms are employed for efficient Bayesian implementation. A detailed exploration of reliability measures—including hazard rate, mean residual life, and stress–strength models—demonstrates the MB distribution’s suitability for complex reliability settings. Extensive Monte Carlo simulations validate the efficiency and precision of the proposed inferential procedures, highlighting significant gains over traditional censoring approaches. Finally, the utility of the methodology is showcased through real-world applications to physics and engineering datasets, where the MB distribution coupled with such censoring yields superior predictive performance. This genuine examination is conducted through two datasets (including the failure times of aircraft windshields, capturing degradation under extreme environmental and operational stress, and mechanical component failure times) that represent recurrent challenges in industrial systems. This work contributes a unified statistical framework that broadens the applicability of the Maxwell–Boltzmann model in reliability contexts and provides practitioners with a powerful tool for decision making under censored data environments.

This paper presents a comprehensive reliability analysis framework for the Chris–Jerry (CJ) lifetime distribution under an improved adaptive progressive Type-II censoring plan. The CJ model, recently introduced to capture skewed lifetime behaviors, is studied under a modified censoring structure designed to provide greater flexibility in terminating life-testing experiments. We derive maximum likelihood estimators for the CJ parameters and key reliability measures, including the reliability and hazard rate functions, and construct approximate confidence intervals using the observed Fisher information matrix and the delta method. To address the intractability of the likelihood function, Bayesian estimators are obtained under independent gamma priors and a squared-error loss function. Because the posterior distributions are not available in closed form, we apply the Metropolis–Hastings algorithm to generate Bayesian estimates and two types of credible intervals. A comprehensive simulation study evaluates the performance of the proposed estimation techniques under various censoring scenarios. The framework is further validated through two real-world datasets: one involving rainfall measurements and another concerning mechanical failure times. In both cases, the CJ model combined with the proposed censoring strategy demonstrates superior fit and reliability inference compared to competing models. These findings highlight the value of the CJ distribution, together with advanced censoring methods, for modeling lifetime data in physics and engineering applications.

This study explores advanced methods for analyzing the two-parameter alpha-power exponential (APE) distribution using data from a novel generalized progressive hybrid censoring scheme. The APE model is inherently asymmetric, exhibiting positive skewness across all valid parameter values due to its right-skewed exponential base, with the alpha-power transformation amplifying or dampening this skewness depending on the power parameter. The proposed censoring design offers new insights into modeling lifetime data that exhibit non-monotonic hazard behaviors. It enhances testing efficiency by simultaneously imposing fixed-time constraints and ensuring a minimum number of failures, thereby improving inference quality over traditional censoring methods. We derive maximum likelihood and Bayesian estimates for the APE distribution parameters and key reliability measures, such as the reliability and hazard rate functions. Bayesian analysis is performed using independent gamma priors under a symmetric squared error loss, implemented via the Metropolis–Hastings algorithm. Interval estimation is addressed using two normality-based asymptotic confidence intervals and two credible intervals obtained through a simulated Markov Chain Monte Carlo procedure. Monte Carlo simulations across various censoring scenarios demonstrate the stable and superior precision of the proposed methods. Optimal censoring patterns are identified based on the observed Fisher information and its inverse. Two real-world case studies—breast cancer remission times and global oil reserve data—illustrate the practical utility of the APE model within the proposed censoring framework. These applications underscore the model’s capability to effectively analyze diverse reliability phenomena, bridging theoretical innovation with empirical relevance in lifetime data analysis.