Progressively Censored Reliability Sampling Plans for the Weibull Distribution

This article presents progressively censored variable sampling plans for the Weibull distribution. Approximate maximum likelihood estimators are developed for estimating the parameters of interest. In the construction of sampling plans, asymptotic distribution theory is used to determine the sample size and the acceptance constant. Sampling plans are tabulated for selected progressive censoring patterns and specifications, for demonstration and comparison. A Monte Carlo experiment, conducted to investigate the accuracy of the asymptotic normal approximation, has shown that the procedure is sufficiently accurate for practical purposes. An example, based on data reported by Montanari and Cacciari from progressively censored aging tests on XLPE-insulated cable models, is given for illustration.     

Inference for two-parameter Rayleigh competing risks data under generalized progressive hybrid censoring

In this paper, a competing risks model based on the generalized progressive hybrid censored two-parameter Rayleigh distributions is studied under the assumption that the lifetime distributions of failure causes are identically distributed with same location and different scale parameters. We obtain maximum likelihood estimates of unknown parameters with associated existence uniqueness. The approximate confidence intervals are constructed using the asymptotic distribution of maximum likelihood estimates via the observed information matrix. Further, Bayes point estimates and the highest probability density credible intervals of unknown parameters are presented, and the Gibbs sampling technique is used to approximate corresponding estimates. A Monte Carlo simulation study is conducted to compare the accuracy of proposed estimates. Finally, a real-life example is presented for illustration purpose.     

Statistical Inference of Chen Distribution Based on Two Progressive Type-II Censoring Schemes

An inverse problem in practical scientific investigations is the process of computing unknown parameters from a set of observations where the observations are only recorded indirectly, such as monitoring and controlling quality in industrial process control. Linear regression can be thought of as linear inverse problems. In other words, the procedure of unknown estimation parameters can be expressed as an inverse problem. However, maximum likelihood provides an unstable solution, and the problem becomes more complicated if unknown parameters are estimated from different samples. Hence, researchers search for better estimates. We study two joint censoring schemes for lifetime products in industrial process monitoring. In practice, this type of data can be collected in fields such as the medical industry and industrial engineering. In this study, statistical inference for the Chen lifetime products is considered and analyzed to estimate underlying parameters. Maximum likelihood and Bayes’ rule are both studied for model parameters. The asymptotic distribution of maximum likelihood estimators and the empirical distributions obtained with Markov chain Monte Carlo algorithms are utilized to build the interval estimators. Theoretical results using tables and figures are adopted through simulation studies and verified in an analysis of the lifetime data. We briefly describe the performance of developed methods.     

Inference for dependence competing risks with partially observed failure causes from bivariate Gompertz distribution under generalized progressive hybrid censoring

Competing risks model is considered with dependence causes of failure in this paper. When the latent failure times are distributed by a bivariate Gompertz model, statistical inference for the unknown model parameters is studied from classical and Bayesian approaches, respectively. Under a generalized progressive hybrid censoring, maximum likelihood estimators of the unknown parameters together with the associated existence and uniqueness are established, and the approximate confidence intervals are also obtained based on asymptotic likelihood theory via the observed Fisher information matrix. Moreover, Bayes estimates and the highest posterior density credible intervals of the unknown parameters are also provided based on a flexible Gamma–Dirichlet prior, and Monte Carlo sampling method is also derived to compute associated estimates. Finally, simulation studies and a real-life example are given for illustration purposes.     

Inference for a geometric‐poisson‐Rayleigh distribution under progressive‐stress accelerated life tests based on type‐I progressive hybrid censoring with binomial removals

Based on failures of a parallel‐series system, a new distribution called geometric‐Poisson‐Rayleigh distribution is proposed. Some properties of the distribution are discussed. A real data set is used to compare the new distribution with other 6 distributions. The progressive‐stress accelerated life tests are considered when the lifetime of an item under use condition is assumed to follow the geometric‐Poisson‐Rayleigh distribution. It is assumed that the scale parameter of the geometric‐Poisson‐Rayleigh distribution satisfies the inverse power law such that the stress is a nonlinear increasing function of time and the cumulative exposure model for the effect of changing stress holds. Based on type‐I progressive hybrid censoring with binomial removals, the maximum likelihood and Bayes (using linear‐exponential and general entropy loss functions) estimation methods are considered to estimate the involved parameters. Some point predictors such as the maximum likelihood, conditional median, best unbiased, and Bayes point predictors for future order statistics are obtained. The Bayes estimates are obtained using Markov chain Monte Carlo algorithm. Finally, a simulation study is performed, and numerical computations are performed to compare the performance of the implemented methods of estimation and prediction.     

Stress‐Strength Reliability Analysis with Extreme Values based on q‐Exponential Distribution

When dealing with practical problems of stress–strength reliability, one can work with fatigue life data and make use of the well‐known relation between stress and cycles until failure. For some materials, this kind of data can involve extremely large values. In this context, this paper discusses the problem of estimating the reliability index R = P(Y < X) for stress–strength reliability, where stress Y and strength X are independent q‐exponential random variables. This choice is based on the q‐exponential distribution's capability to model data with extremely large values. We develop the maximum likelihood estimator for the index R and analyze its behavior by means of simulated experiments. Moreover, confidence intervals are developed based on parametric and nonparametric bootstrap. The proposed approach is applied to two case studies involving experimental data: The first one is related to the analysis of high‐cycle fatigue of ductile cast iron, whereas the second one evaluates the specimen size effects on gigacycle fatigue properties of high‐strength steel. The adequacy of the q‐exponential distribution for both case studies and the point and interval estimates based on maximum likelihood estimator of the index R are provided. A comparison between the q‐exponential and both Weibull and exponential distributions shows that the q‐exponential distribution presents better results for fitting both stress and strength experimental data as well as for the estimated R index. Copyright © 2016 John Wiley & Sons, Ltd.     

Classical and Bayesian inference on progressive‐stress accelerated life testing for the extension of the exponential distribution under progressive type‐II censoring

In this paper, a progressive‐stress accelerated life test under progressive type‐II censoring is considered. The cumulative exposure model is assumed when the lifetime of test units follows an extension of the exponential distribution. The maximum likelihood and Bayes estimates of the model parameters are obtained. The approximate and credible confidence intervals of the estimators are derived. Furthermore, a real lifetime data set is analyzed to illustrate the proposed procedures. Finally, the simulation studies are used to compare between 2 different designs of the progressive‐stress test (simple and multiple ramp‐stress tests).     

Optimal design of a simple step‐stress accelerated life test under progressive type I censoring with nonuniform durations for exponential lifetimes

Thanks to continuously advancing technology and manufacturing processes, the products and devices are becoming highly reliable. However, performing the life tests of these products at normal operating conditions becomes extremely difficult, if not impossible, due to their long life spans. This can result in missed opportunities to introduce the products to the market in a timely manner and eventually loss of the market share. This problem is solved by accelerated life tests where the test units are subjected to higher stress levels than the normal usage level so that information on the lifetime parameters can be obtained more quickly. The lifetime at the design condition is then estimated through extrapolation using a regression model. In this work, the design optimization of a simple step‐stress accelerated life test under progressive type I censoring is studied with nonuniform step durations for assessing the reliability characteristics of a solar lighting device. Allowing the intermediate censoring to take place at the stress change time point, the nature of the optimal stress duration is demonstrated under various design criteria including D‐optimality, C‐optimality, A‐optimality, and E‐optimality. The existence of these optimal designs is investigated in detail for exponential lifetimes with a single stress variable, and the effect of the intermediate censoring proportion is assessed on the design efficiency.     

35CrMoV钢硬度概率分布与抗拉强度设计许用值预测分析

采用最小二乘法和K-S检验方法进行分布拟合检验,在求得35CrMoV钢硬度和抗拉强度正态分布的基础上,对其硬度和抗拉强度关系进行相关性研究,运用线性相关系数判断硬度和抗拉强度之间具有强正线性相关关系.运用正态代数运算方法,从硬度的概率分布估计抗拉强度的概率分布,预测其设计许用值,并与从抗拉强度概率分布计算得到的设计许用...