Search Results (60)

This study focuses on estimating the unknown parameters and the reliability function of the inverted-Weibull distribution, using an improved adaptive progressive Type-II censoring scheme under a competing risks model. Both classical and Bayesian estimation approaches are explored to offer a thorough analysis. Under the classical approach, maximum likelihood estimators are obtained for the unknown parameters and the reliability function. Approximate confidence intervals are also constructed to assess the uncertainty in the estimates. From a Bayesian standpoint, symmetric Bayes estimates and highest posterior density credible intervals are computed using Markov Chain Monte Carlo sampling, assuming a symmetric squared error loss function. An extensive simulation study is carried out to assess how well the proposed methods perform under different experimental conditions, showing promising accuracy. To demonstrate the practical use of these methods, a real dataset is analyzed, consisting of the survival times of male mice aged 35 to 42 days after being exposed to 300 roentgens of X-ray radiation. The analysis demonstrated that the inverted Weibull distribution is well-suited for modeling the given dataset. Furthermore, the Bayesian estimation method, considering both point estimates and interval estimates, was found to be more effective than the classical approach in estimating the model parameters as well as the reliability function. Full article

This study offers a newly improved Type-II adaptive progressive censoring with data sampled from an inverse XLindley (IXL) distribution for more efficient and adaptive reliability assessments. Through this sampling mechanism, we evaluate the parameters of the IXL distribution, as well as its reliability and hazard rate features. In the context of reliability, to handle flexible and time-constrained testing frameworks in high-reliability environments, we formulate maximum likelihood estimators versus Bayesian estimates derived via Markov chain Monte Carlo techniques under gamma priors, which effectively capture prior knowledge. Two patterns of asymptotic interval estimates are constructed through the normal approximation of the classical estimates and of the log-transformed classical estimates. On the other hand, from the Markovian chains, two patterns of credible interval estimates are also constructed. A robust simulation study is carried out to compare the classical and Bayesian point estimation methods, along with the four interval estimation methods. This study’s practical usefulness is demonstrated by its analysis of a real-world dataset. The results reveal that both conventional and Bayesian inferential methods function accurately, with the Bayesian outcomes surpassing those of the conventional method. Full article

In recent years, there has been an increasing interest in the application of progressive censoring as a means to reduce both cost and experiment duration. In the absence of explanatory variables, the present study employs a statistical inference approach for the inverse Weibull distribution, using a progressive type II censoring strategy with two independent samples. The article expounds on the maximum likelihood estimation method, utilizing the Fisher information matrix to derive approximate confidence intervals. Moreover, interval estimations are computed by the bootstrap method. We explore the application of Bayesian methods for estimating model parameters under both the squared error and LINEX loss functions. The Bayesian estimates and corresponding credible intervals are calculated via Markov chain Monte Carlo (MCMC). Finally, comprehensive simulation studies and real data analysis are carried out to validate the precision of the proposed estimation methods. Full article

Traditional reliability methods using fixed removal plans often overlook withdrawal randomness, leading to biased estimates for asymmetric data. This study advances classical and Bayesian frameworks for the inverse Chen distribution, which is suited for modeling asymmetric data under adaptive progressively Type-II censoring with binomial removals. Here, removals post-failure follow a dynamic binomial process, enhancing a more realistic approach for reliability studies. Maximum likelihood estimates are computed numerically, with confidence intervals derived asymptotically. Bayesian approaches employ gamma priors, symmetric squared error loss, and posterior sampling for estimates and credible intervals. A simulation study validates the methods, while two asymmetric real-world applications demonstrate practicality: (1) analyzing diamond sizes from South-West Africa, capturing skewed geological distributions, and (2) modeling failure times of airborne communication transceivers, vital for aviation safety. The flexibility of the inverse Chen in handling asymmetric data addresses the limitations of symmetric assumptions, offering precise reliability tools for complex scenarios. This integration of adaptive censoring and asymmetric distributions advances reliability analysis, providing robust solutions where traditional approaches falter. Full article

This study introduces an objective Bayesian approach for estimating the reliability of a multicomponent stress–strength model based on the Pareto distribution under an adaptive progressive Type-II censoring scheme. The proposed method is developed within a Bayesian framework, utilizing a reference prior with partial information to improve the accuracy of point estimation and to ensure the construction of a credible interval for uncertainty assessment. This approach is particularly useful for addressing several limitations of a widely used likelihood-based approach in estimating the multicomponent stress–strength reliability under the Pareto distribution. For instance, in the likelihood-based method, the asymptotic variance–covariance matrix may not exist due to certain constraints. This limitation hinders the construction of an approximate confidence interval for assessing the uncertainty. Moreover, even when an approximate confidence interval is obtained, it may fail to achieve nominal coverage levels in small sample scenarios. Unlike the likelihood-based method, the proposed method provides an efficient estimator across various criteria and constructs a valid credible interval, even with small sample sizes. Extensive simulation studies confirm that the proposed method yields reliable and accurate inference across various censoring scenarios, and a real data application validates its practical utility. These results demonstrate that the proposed method is an effective alternative to the likelihood-based method for reliability inference in the multicomponent stress–strength model based on the Pareto distribution under an adaptive progressive Type-II censoring scheme. Full article

This study addresses the problem of parameter estimation and prediction for type-II censored data from the two-parameter Birnbaum–Saunders (BS) distribution. The BS distribution is commonly used in reliability analysis, particularly in modeling fatigue life. Accurate estimation and prediction are crucial in many fields where censored data frequently appear, such as material science, medical studies and industrial applications. This paper presents both frequentist and Bayesian approaches to estimate the shape and scale parameters of the BS distribution, along with the prediction of unobserved failure times. Random data are generated from the BS distribution under type-II censoring, where a pre-specified number of failures (m) is observed. The generated data are used to calculate the Maximum Likelihood Estimation (MLE) and Bayesian inference and evaluate their performances. The Bayesian method employs Markov Chain Monte Carlo (MCMC) sampling for point predictions and credible intervals. We apply the methods to both datasets generated under type-II censoring and real-world data on the fatigue life of 6061-T6 aluminum coupons. Although the results show that the two methods yield similar parameter estimates, the Bayesian approach offers more flexible and reliable prediction intervals. Extensive R codes are used to explain the practical application of these methods. Our findings confirm the advantages of Bayesian inference in handling censored data, especially when prior information is available for estimation. This work not only supports the theoretical understanding of the BS distribution under type-II censoring but also provides practical tools for analyzing real data in reliability and survival studies. Future research will discuss extensions of these methods to the multi-sample progressive censoring model with larger datasets and the integration of degradation models commonly encountered in industrial applications. Full article

The Bivariate Odd Lindley Half-Logistic (BOLiHL) distribution with progressive Type-II censoring provides a powerful statistical tool for analyzing dependent data effectively. This approach benefits society by enhancing engineering systems, improving healthcare decisions, and supporting effective risk management, all while optimizing resources and minimizing experimental burdens. In this paper, the likelihood function derived under progressive Type-II censoring is generalized for the BOLiHL distribution. The well-known maximum likelihood estimation method and Bayesian estimation are applied to evaluate the parameters of the distribution. A study utilizing simulation techniques is performed to evaluate the performance of the estimators, using statistical analysis metrics for censored observations under a progressive Type-II censoring scheme with varying sample sizes, failure times, and censoring schemes. Additionally, a real dataset is studied to validate the proposed model, delivering impactful analyses for practical applications. Full article

This study explores accelerated life tests to examine the durability of highly reliable products. These tests involve applying higher stress levels, such as increased temperature, voltage, or pressure, that cause early failures. The power half-logistic (PHL) distribution is utilized due to its flexibility in modeling the probability density and hazard rate functions, effectively representing various data patterns commonly encountered in practical applications. The step stress partially accelerated life testing model is analyzed under an adaptive type II progressive censoring scheme, with samples drawn from the PHL distribution. The maximum likelihood method estimates model parameters and calculates asymptotic confidence intervals. Bayesian estimates are also obtained using Lindley’s approximation and the Markov Chain Monte Carlo (MCMC) method under different loss functions. Additionally, D- and A-optimality criteria are applied to determine the optimal stress-changing time. Simulation studies are conducted to evaluate the performance of the estimation methods and the optimality criteria. Finally, real-world datasets are analyzed to demonstrate the practical usefulness of the proposed model. Full article

The aim of this research is to estimate the parameters of the modified Frechet-exponential (MFE) distribution using different methods when applied to progressive type-II censored samples. These methods include using the maximum likelihood technique and the Bayesian approach, which were used to determine the values of parameters in addition to calculating the reliability and failure functions at time t. The approximate confidence intervals (ACIs) and credible intervals (CRIs) are derived for these parameters. Two bootstrap techniques of parametric type are provided to compute the bootstrap confidence intervals. Both symmetric loss functions such as the squared error loss (SEL) and asymmetric loss functions such as the linear-exponential (LINEX) loss are used in the Bayesian method to obtain the estimates. The Markov Chain Monte Carlo (MCMC) technique is utilized in the Metropolis–Hasting sampler approach to obtain the unknown parameters using the Bayes approach. Two actual datasets are utilized to examine the various progressive schemes and different estimation methods considered in this paper. Additionally, a simulation study is performed to compare the schemes and estimation techniques. Full article

The lifetime performance index (LPI) is an important metric for evaluating product quality, and research on the statistical inference of the LPI is of great significance. This paper discusses both the classical and Bayesian estimations of the LPI under an adaptive progressive type-II censored lifetime test, assuming that the product’s lifetime follows a generalized inverse Lindley distribution. At first, the maximum likelihood estimator of the LPI is derived, and the Newton–Raphson iterative method is adopted to solve the numerical solution due to the log-likelihood equations having no analytical solutions. If the exact distribution of the LPI is not available, then the asymptotic confidence interval and bootstrap confidence interval of the LPI are constructed. For the Bayesian estimation, the Bayesian estimators of the LPI are derived under three different loss functions. Due to the complex multiple integrals involved in these estimators, the MCMC method is used to draw samples and further construct the HPD credible interval of the LPI. Finally, Monte Carlo simulations are used to observe the performance of these estimators in terms of the average bias and mean squared error, and two practical examples are used to illustrate the application of the proposed estimation method. Full article

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. Full article

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. Full article

Abstract: This paper is concerned with applying the Bayesian and E-Bayesian approaches to estimating the unknown parameters of the modified Topp–Leone–Chen distribution under a progressive Type-II censored sample plan. The paper explores the complexities of different estimating methods and investigates the behavior of the estimates through some computations. The Bayes and E-Bayes estimators are obtained under two distinct loss functions, the balanced squared error loss function, as a symmetric loss function, and the balanced linear exponential loss function, as an asymmetric loss function. The estimators are derived using gamma prior and uniform hyperprior distributions. A numerical illustration is given to examine the theoretical results through using the Metropolis–Hastings algorithm of the Markov chain Monte Carlo method of simulation by the R programming language. Finally, real-life data sets are applied to prove the flexibility and applicability of the model. Full article

This article discusses different methods for estimating the shape and scale parameters of the generalized inverted exponential distribution (GIED) and the acceleration factor in constant stress partially accelerated life test (CSPALT) with general progressively Type-II censored samples. We obtain the maximum likelihood estimates for the three parameters and calculate correlated approximate confidence intervals. Bayesian point estimates and credible intervals are also determined using the importance sampling method. Monte-Carlo simulation studies are conducted to demonstrate and compare the effectiveness of the proposed parameter estimation techniques. Additionally, a real-life dataset is examined to highlight the practical utility of these methodologies. Our findings indicate that the GIED provides an appropriate and flexible model for the real lifetime data, and the Bayesian approach offers better estimation than classical methods under most scenarios, in terms of using generally progressively Type-II censored samples under CSPALT. Full article

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples. Full article