Search Results (50)

This study examines likelihood-based estimation of the joint survival function $S(t_1,t_2)=\mathrm{Pr}\{T^{\left(1\right)}>t_1,T^{\left(2\right)}>t_2\}$ for systems with two competing failure modes observed under right censoring. Rather than introducing a new distributional family, the study compares established dependence mechanisms within a common observed-data framework. Exponential and Weibull margins are combined with three types of dependence: Archimedean copulas, represented by the Gumbel and Clayton families; shared gamma frailty, used to model latent measurement-level heterogeneity; and Marshall–Olkin extensions, used to represent common shocks and simultaneous failures. The same observation scheme, likelihood construction, censoring design, and performance criteria are used across models. Model performance is evaluated through Monte Carlo simulation using bias, integrated mean squared error, and empirical coverage, and the workflow is illustrated with the Device G reliability data. The results show that ignoring dependence can distort joint survival estimates, especially under moderate or high censoring. They also show that copula, frailty, and Marshall–Olkin specifications can lead to different reliability assessments because they encode different stochastic mechanisms. The estimation workflow includes multi-start optimization and diagnostics for boundary solutions, Hessian stability, and irregular likelihood behavior. Full article

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. Full article

In this paper, we introduce a new extension of the Topp–Leone-G family, called the generalized Marshall–Olkin Topp–Leone-G (GMOTL-G) family of distributions. The proposed family is obtained by combining the generalized Marshall–Olkin and Topp–Leone-G generators, leading to a more flexible class of models for lifetime data. We study several of its mathematical and statistical properties and focus in particular on the generalized Marshall–Olkin Topp–Leone exponential (GMOTL-E) distribution as an important special case. For this model, we derive and discuss a number of useful characteristics, including the moment generating function, moments, order statistics, residual and reversed residual life functions, mean deviations, asymptotic behavior, and stochastic ordering. We also develop maximum likelihood estimation for the model parameters under both complete and right-censored samples. In addition, we construct a goodness-of-fit test for the proposed model under independent right censoring using a chi-square type approach. The performance of the estimation and testing procedures is investigated through simulation, and the results show good behavior of the estimators and satisfactory agreement between empirical and theoretical significance levels. Finally, two real data applications, one with complete data and one with right-censored data, are presented to illustrate the flexibility and practical usefulness of the proposed model. These results show that the new family provides an effective tool for modeling lifetime data and for assessing model adequacy in the presence of right censoring. Full article

The Marshall–Olkin family of distributions has gained increasing attention in fields such as reliability engineering, survival analysis, financial risk modeling, and actuarial science because of its flexibility in modeling dependence among events and its wide range of extensions. Despite its growing relevance, a systematic understanding of how research on Marshall–Olkin models has evolved over time is still limited. This study addresses this gap by combining bibliometric techniques with topic modeling to analyze the structure and evolution of the scientific literature on Marshall–Olkin models. The analysis includes all 266 peer-reviewed publications on Marshall–Olkin models indexed in Scopus between 1981 and 2025. Bibliometric techniques (including heatmaps, clustering analyses, and temporal visualizations) are used to characterize publication patterns, source relationships, and thematic evolution. In addition, Latent Dirichlet Allocation (LDA) uncovered 27 topics and examined their prevalence across journals and time periods. The results reveal five main clusters of publication sources and three temporal groupings derived from hierarchical clustering of topic distributions, reflecting the thematic progression of the field. Overall, the findings highlight both the persistence of core research themes and the emergence of new applications, particularly in areas such as Bayesian competing risks, censoring models, and parameter estimation in Weibull-based frameworks. This study provides a systematic and data-driven perspective on the intellectual evolution of Marshall–Olkin research, helping scholars identify emerging trends and potential directions for future work. Full article

The prediction of material lifetime is central to nanomaterial engineering and reliability analysis. We propose the Marshall–Olkin Power Half-Logistic (MOPHL) distribution, obtained by applying a Marshall–Olkin transform to the Power Half-Logistic baseline. We derive core properties—including moments, hazard rate characterization, and Rényi entropy—and develop inference under generalized progressive hybrid censoring. Estimation is carried out via maximum likelihood and Bayesian methods using a Metropolis–Hastings sampler. Asymptotic results, Fisher information, and corresponding confidence/credible intervals are provided. A Monte Carlo study assesses bias, the mean squared error, and coverage across censoring scenarios and hazard regimes. In a case study on hydroxylated fullerene degradation, MOPHL outperforms nine competing models in goodness-of-fit and predictive reliability. We also report the mean time to failure and mean residual life to support engineering decision-making. The proposed framework offers a tractable and robust tool for degradation analysis under censored data, with applicability to materials, mechanical components, biomedical devices, and environmental monitoring. Full article

This study introduces a novel four-parameter lifetime distribution constructed within the Topp–Leone Power Gompertz framework. Owing to its flexible structure, the proposed model accommodates a wide range of density shapes and hazard-rate patterns, including increasing, decreasing, bathtub-shaped, unimodal, and other non-monotone behaviors. Key distributional properties, including moments, entropy-based measures, quantile-based measures, and order statistics, are derived. Parameter inference is conducted using both likelihood-based and Bayesian approaches, and the finite-sample performance of the resulting estimators is assessed via Monte Carlo simulations. The practical relevance of the proposed distribution is illustrated using two real datasets and benchmarked against several competing lifetime models, including the Gompertz, Power Gompertz, Weibull, Topp–Leone Gompertz, Marshall–Olkin Gompertz, and Exponentiated Gompertz distributions. Overall, the comparative analyses demonstrate the superior fitting performance of the proposed model, highlighting its effectiveness for complex reliability, survival, and pharmacokinetic data. Full article

This paper introduces a novel bivariate distribution, referred to as the Bivariate Burr XII Inverse Weibull (BBXII-IW) distribution, constructed via the Marshall–Olkin approach from the univariate Burr XII Inverse Weibull (BXII-IW) distribution. The proposed BBXII-IW model provides a flexible framework for modeling dependent bivariate data, including competing risk scenarios. The key statistical properties of the distribution are derived, and parameter estimation is conducted using the maximum likelihood method. The model’s performance is evaluated using two types of real-world datasets: (1) bivariate data and (2) dependent competing risk data related to diabetic retinopathy. The results demonstrate that the BBXII-IW distribution offers an improved fit compared to existing models, highlighting its flexibility and practical relevance in modeling complex dependent structures. Full article

This paper introduces the Marshall–Olkin unit-exponentiated-half-logistic (MO-UEHL) distribution, a novel three-parameter model designed to enhance the flexibility of the unit-exponentiated-half-logistic distribution through the incorporation of the Marshall–Olkin transformation. Defined on the unit interval $(0,1)$, the MO-UEHL distribution is well-suited for modeling proportional data exhibiting asymmetry. The Marshall–Olkin tilt parameter $\alpha$ explicitly controls the degree and direction of asymmetry, enabling the density to range from highly right-skewed to nearly symmetric unimodal forms, and even to left-skewed configurations for certain parameter values, thereby offering a direct mathematical representation of symmetry breaking in bounded proportional data. The resulting model achieves this versatility without relying on exponential terms or special functions, thus simplifying computational procedures. We derive its key mathematical properties, including the probability density function, cumulative distribution function, survival function, hazard rate function, quantile function, moments, and information-theoretic measures such as the Shannon and residual entropy. Parameter estimation is explored using maximum likelihood, maximum product spacing, ordinary and weighted least-squares, and Cramér–von Mises methods, with simulation studies evaluating their performance across varying sample sizes and parameter sets. The practical utility of the MO-UEHL distribution is demonstrated through applications to four real datasets from environmental and engineering contexts. The results highlight the MO-UEHL distribution’s potential as a valuable tool in reliability analysis, environmental modeling, and related fields. Full article

The Gompertz distribution has long been a cornerstone for analyzing growth processes and mortality patterns across various scientific disciplines. However, as the intricacies of real-world phenomena evolve, there is a pressing need for more versatile probability distributions that can accurately capture a wide array of data characteristics. In response to this demand, we introduce the Marshall–Olkin Power Gompertz (MOPG) distribution, an innovative and powerful extension of the traditional Gompertz model. The MOPG distribution is crafted by enhancing the Power Gompertz cumulative distribution function through the Marshall–Olkin transformation. This distribution yields two pivotal contributions: a power parameter (c) that significantly increases the model’s adaptability to diverse data patterns and the Marshall–Olkin transformation, which modifies tail behavior to enhance predictive accuracy. Furthermore, we derived the distribution’s essential statistical properties and evaluate its performance through extensive Monte Carlo simulations, along with a maximum likelihood estimation of model parameters. Our empirical validation, utilizing three real-world data sets, compellingly demonstrated that the MOPG distribution not only surpasses several well-established lifetime distributions but is also superior in terms of flexibility and tail behavior characterization. The results highlight that the proposed MOPG stands out as a superior choice, delivering the most precise fit to the data when compared to various competing models, and its performance makes it a compelling option worth considering. Full article

In this paper, a new generalization of continuous distributions using the Marshall–Olkin distribution as a generator is proposed and studied. Many important mathematical properties are derived from the proposed distribution, including moments, the moment generating function, order statistics, entropy, and the quantile function. Two different estimation methods are used, namely, maximum likelihood estimation and Bayesian methods. A Monte Carlo simulation is conducted to estimate the parameters and study the behavior of the proposed distribution. Bayesian estimation is obtained using the Gibbs sampler and Metropolis–Hastings algorithm. Finally, two real-world datasets are used to compare the performance of non-Bayesian and Bayesian methods. Full article

We develop a novel family of distributions named the Marshall–Olkin type II exponentiated half-logistic–odd Burr X-G distribution. Several mathematical properties including linear representation of the density function, Rényi entropy, probability-weighted moments, and distribution of order statistics are obtained. Different estimation methods are employed to estimate the unknown parameters of the new distribution. A simulation study is conducted to assess the effectiveness of the estimation methods. A special model of the new distribution is used to show its usefulness in various disciplines. Full article

This paper explores a progressive-stress accelerated life test under progressive type-II censoring with binomial random removal. It assumes a cumulative exposure model in which the lifetimes of test units follow a Marshall–Olkin length-biased exponential distribution. The study derives maximum likelihood and Bayes estimates of the model parameters and constructs Bayes estimates of the unknown parameters under various loss functions. In addition, this study provides approximate, credible, and bootstrapping confidence intervals for the estimators. Moreover, it evaluates three optimal test methods to determine the most effective censoring approach based on various optimality criteria. A real-life dataset is analyzed to demonstrate the proposed procedures and simulation studies used to compare two different designs of the progressive-stress test. Full article

This paper focuses on applying the Marshall-Olkin approach to generate a new bivariate distribution. The distribution is called the bivariate exponentiated Lomax distribution, and its marginal distribution is the exponentiated Lomax distribution. Numerous attributes are examined, including the joint reliability and hazard functions, the bivariate probability density function, and its marginals. The joint probability density function and joint cumulative distribution function can be stated analytically. Different contour plots of the joint probability density function and joint reliability and hazard rate functions of the bivariate exponentiated Lomax distribution are given. The unknown parameters and reliability and hazard rate functions of the bivariate exponentiated Lomax distribution are estimated using the maximum likelihood method. Also, the Bayesian technique is applied to derive the Bayes estimators and reliability and hazard rate functions of the bivariate exponentiated Lomax distribution. In addition, maximum likelihood and Bayesian two-sample prediction are considered to predict a future observation from a future sample of the bivariate exponentiated Lomax distribution. A simulation study is presented to investigate the theoretical findings derived in this paper and to evaluate the performance of the maximum likelihood and Bayes estimates and predictors. Furthermore, the real data set used in this paper comprises the scoring times from 42 American Football League matches that took place over three consecutive independent weekends in 1986. The results of utilizing the real data set approve the practicality and flexibility of the bivariate exponentiated Lomax distribution in real-world situations, and the bivariate exponentiated Lomax distribution is suitable for modeling this bivariate data set. Full article

In this paper, we generalize two new statistical distributions, to improve the ability to model failure rates with non-monotonic, monotonic, and mainly bathtub curve behaviors. We call these distributions Generalized Powered Uniform Distribution and MOE-Powered Uniform. The proposed distributions’ approach is based on incorporating a parameter k in the power of the values of the random variables, which is associated with the Probability Density Function and includes an operator called the Powered Mean. Various statistical and mathematical features focused on reliability analysis are presented and discussed, to make the models attractive to reliability engineering or medicine specialists. We employed the Maximum Likelihood Estimator method to estimate the model parameters and we analyzed its performance through a Monte Carlo simulation study. To demonstrate the flexibility of the proposed approach, a comparative analysis was carried out on four case studies with the proposed MOE-Powered Uniform distribution, which can model failure times as a bathtub curve. The results showed that this new model is more flexible and useful for performing reliability analysis. Full article

We propose the “Marshall–Olkin Bivariate Weibull Model with Modified Singularity MOBW-$\mu$”, which focuses on bivariate distributions essential for reliability and survival analyses. Distributions such as the Marshall–Olkin bivariate exponential (MOBE) and the Marshall–Olkin bivariate Weibull (MOBW) are discussed. The MOBW-$\mu$ model is introduced, which incorporates a lag parameter $\mu$ in the singular part, and probabilistic properties such as the joint survival function, marginal density functions, and the bivariate hazard rate function are explored. In addition, aspects such as the correlation structure and survival copulation are addressed and we show that the correlation of the MOBW-$\mu$ is always lower than that of its copula, regardless of the parameters. The latter result implies that the MOBW-$\mu$ does not have the Lancaster’s phenomenon that explains that any nonlinear transformation of variables decreases the correlation in absolute value. The article concludes by presenting a robust theoretical framework applicable to various disciplines. Full article