Generalized inverted exponential distribution under constant stress accelerated life test: Different estimation methods with application

Estimation of the parameters of generalized inverted exponential distribution is considered under constant stress accelerated life test. Besides the maximum likelihood method, nine different frequentist methods of estimation are used to estimate the unknown parameters. Moreover, the reliability function is estimated under use conditions based on different methods of estimation. We perform extensive simulation experiments to see the performance of the proposed estimators. As an illustration, a real data set is analyzed to demonstrate how the proposed methods may work in practice.     

Different Estimation Methods for Constant Stress Accelerated Life Test under the Family of the Exponentiated Distributions

The Accelerated Life Testing (ALT) has been used for a long time in several fields to obtain information on the reliability of product components and materials under operating conditions in a much shorter time. One of the main purposes of applying ALT is to estimate the failure time functions and reliability performance under normal conditions. This paper concentrates on the estimation procedures under ALT and how to select the best estimation method that gives accurate estimates for the reliability function. For this purpose, different estimation methods are used, such as maximum likelihood, least squares (LS), weighted LS, and probability weighted moment. Moreover, the reliability function under usual conditions is predicted. The estimation procedures are applied under the family of the exponentiated distributions in general, and for the exponentiated inverted Weibull (EIW) as a special case. Numerical analysis including simulated data and a real life data set is conducted to compare the performances between these four methods. It is found that the ML method gives the best results among other estimation methods. Finally, a comparison between the EIW and the Inverted Weibull (IW) distributions based on a real life data set is made using a likelihood ratio test. It is observed that the EIW distribution can provide better fitting than the IW in case of ALT. Copyright © 2015 John Wiley & Sons, Ltd.     

Reliability analysis of Weibull multicomponent system with stress-dependent parameters from accelerated life data

In this paper, reliability estimation of multicomponent system under a multilevel accelerated life testing. When the lifetime of components follows Weibull distribution, the problem of point and interval estimates are discussed from different perspectives. Under a general life-stress assumption that there are multiple nonconstant and stress-dependent scale and shape parameters, the maximum likelihood estimates of unknown parameters along with associated existence and uniqueness are established. Approximate confidence intervals are constructed correspondingly via expected Fisher information matrix. Furthermore, some pivotal quantities are constructed and alternative generalized point and interval estimates are also proposed for comparison. In addition, predictive intervals for the lifetime of the multicomponent system are discussed under classical and generalized pivotal approaches, respectively. The results show that the proposed generalized estimates are superior to the conventional likelihood approach in terms of the accuracy. A real data example is carried out to illustrate the implementations of the proposed methods.     

A sequential constant‐stress accelerated life testing scheme and its Bayesian inference

In the analysis of accelerated life testing (ALT) data, some stress‐life model is typically used to relate results obtained at stressed conditions to those at use condition. For example, the Arrhenius model has been widely used for accelerated testing involving high temperature. Motivated by the fact that some prior knowledge of particular model parameters is usually available, this paper proposes a sequential constant‐stress ALT scheme and its Bayesian inference. Under this scheme, test at the highest stress is firstly conducted to quickly generate failures. Then, using the proposed Bayesian inference method, information obtained at the highest stress is used to construct prior distributions for data analysis at lower stress levels. In this paper, two frameworks of the Bayesian inference method are presented, namely, the all‐at‐one prior distribution construction and the full sequential prior distribution construction. Assuming Weibull failure times, we (1) derive the closed‐form expression for estimating the smallest extreme value location parameter at each stress level, (2) compare the performance of the proposed Bayesian inference with that of MLE by simulations, and (3) assess the risk of including empirical engineering knowledge into ALT data analysis under the proposed framework. Step‐by‐step illustrations of both frameworks are presented using a real‐life ALT data set. Copyright © 2008 John Wiley & Sons, Ltd.     

Statistical inference on progressive-stress accelerated life testing for the logistic exponential distribution under progressive type-II censoring

The accelerated life testing (ALT) is an efficient approach and has been used in several fields to obtain failure time data of test units in a much shorter time than testing at normal operating conditions. In this article, a progressive-stress ALT under progressive type-II censoring is considered when the lifetime of test units follows logistic exponential distribution. We assume that the scale parameter of the distribution satisfying the inverse power law. First, the maximum likelihood estimates of the model parameters and their approximate confidence intervals are obtained. Next, we obtain Bayes estimators under squared error loss function with the help of Metropolis-Hasting (MH) algorithm. We also derive highest posterior density (HPD) credible intervals of the model parameters. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation. Finally, one data set has been analyzed for illustrative purposes.     

Robustness of asymptotic accelerated life test plans to small‐sample settings

Much of the statistical literature on optimal test planning for accelerated life testing utilize asymptotic methods to derive optimal test plans. While sufficient effort is made to assess the robustness of these test plans to the choice of design parameters and distribution assumptions, there is very little literature on the performance of asymptotic test plans relative to small samples (on the order of 10‐15 samples). An alternative concern is that the asymptotic test plans may not necessarily be the true “optimal” test plan for a given sample size. The purpose of this research is to present exact or “near‐exact” methods for developing test plans and compare the performance of these test plans with corresponding asymptotic test plans in small‐sample settings. The optimal location of design points and sample allocation is determined using each method for lognormal and Weibull lifetime distributions with both complete and Type 1 right‐censored data under two selected acceleration factor models. The investigations reveal that asymptotic test plans tend to corroborate quite well with exact test plans and thus are suitably robust to small‐sample settings in terms of optimal variance.     

A Bayesian generalized Eyring-Weibull accelerated life testing model

In this paper, a novel approach to a Bayesian accelerated life testing model is presented. The Weibull distribution is used as the life distribution and the generalized Eyring model as the time transformation function. This is a model that allows for the use of more than one stressor, whereas other commonly used acceleration models, such as the Arrhenius and power law models, incorporate one stressor. The use of the generalized Eyring-Weibull model developed in this paper is demonstrated in a case study, where Markov chain Monte Carlo methods are utilized to generate samples for posterior inference.     

Optimum step‐stress for temperature accelerated life testing

Step‐stress accelerated life testing is a design strategy where the stress is modified several times during the test. In this work we address the problem of designing such a test. We focus on temperature accelerated life testing and we address the problems of setting the step duration and the stress levels. Assuming an Arrhenius model, maximum likelihood estimates of the parameters are computed. Relying on the properties of these estimators we compare different criteria for assessing the optimality of the plans produced. Some tables are presented to illustrate the method. For a fixed number of steps and a set of temperatures, a table of optimal length steps can be computed. For fixed step lengths, sets of temperatures leading to optimal plans are also available. Thus, this work provides useful tools to help engineers make decisions in testing strategy. Copyright © 2007 John Wiley & Sons, Ltd.     

An inference method for temperature step‐stress accelerated life testing

This paper deals with step‐stress accelerated life testing. It presents a practical method to analyse temperature step‐stress accelerated life test data. The Arrhenius model is considered. Activation energy and failure rate under operational conditions are estimated both graphically and using maximum likelihood. Applications on simulated data and on real data are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Validating circuit board Interconnect Stress Test preconditioning processes using statistical model comparisons of accelerated test data

Interconnect stress testing (IST) is used for acceptance of product by testing coupons fabricated along with the production panels to determine the long‐term reliability of the plated thru‐holes. Prior to in‐service use, the actual production circuit boards are subjected to Circuit Card Assembly processes like soldering components, but simulated on coupons using preconditioning cycles (PCCs). In this paper, the Circuit Card Assembly preconditioning process is compared with IST PCCs. Accelerated test data for this study use IST preconditioning and Re‐Flow Oven preconditioning at an independent company laboratory from coupons supplied by three fabrication suppliers. The data indicate that a Weibull distribution be applied for statistical model comparisons using the likelihood‐ratio test on both types of preconditioning processes with coupons that are then subjected to subsequent in‐service use IST cycles until failure occurs. Results show that using six IST PCCs are statistically equivalent to six Re‐Flow Oven PCCs and thus five IST PCCs currently in use should be reconsidered, but the advantages of effective IST preconditioning lower acceptance costs significantly. In addition, the statistical results are validated by root mean square power calculations. Copyright © 2009 John Wiley & Sons, Ltd.     

A fatigue model with local sensitivity analysis

The goal of this paper is two fold. First, it introduces a general parametric lifetime model for high‐cycle fatigue regime derived from physical, statistical, engineering and dimensional analysis considerations. The proposed model has two threshold parameters and three Weibull distribution parameters. A two‐step procedure is presented to estimate the parameters. In the first step, the two threshold parameters are estimated by minimizing a least squares regression function. In the second step, the parameters are estimated by the maximum likelihood method after pooling together the data from different stress levels. Since parameter estimation should always be accompanied by a sensitivity analysis of the fitted model, the second goal of this paper is to propose a method for sensitivity analysis for fatigue models. We show that the proposed sensitivity analysis methods are general and can be applied to any fatigue or lifetime model, not just to the one proposed here. Although several fatigue models have been proposed in the literature, to our knowledge this is the first attempt to produce methods for sensitivity analysis for fatigue models. The proposed method makes use of the well‐known duality property of mathematical programming, which states that the partial derivatives of the primal objective function with respect to the constraints right hand side parameters are the optimal values of the negative of the dual problem variables. For the parameters or data, for which sensitivities are sought, to appear on the right hand side, they are converted into artificial variables and set to their actual values, thus obtaining the desired constraints. Both the estimation and sensitivity analysis methods are illustrated by two examples, one application using real fatigue data and the other using simulated data. In addition, the sensitivity proposed method is also applied to an alternative fatigue model. Finally, some specific conclusions and recommendations are also given.     

Reliability analysis of the dynamic system for the Chen model through sequential order statistics

Recently, the two-parameter Chen distribution has widely been used for reliability studies in various engineering fields. In this article, we have developed various statistical inferences on the composite dynamic system, assuming Chen distribution as a baseline model. In this dynamic system, failure of a component induces a higher load on the surviving components and thus increases component hazard rate through a power-trend process. The classical and Bayesian point estimates of the unknown parameters of the composite system are obtained by the method of maximum likelihood and Markov chain Monte Carlo techniques, respectively. In the Bayesian framework, we have used gamma priors to obtain Bayes estimates of unknown parameters under the squared error and generalized entropy loss functions. The interval estimates of the baseline reliability function are obtained by using the Fisher information matrix and Bayesian method. A parametric hypothesis test is presented to test whether the failed components change the hazard rate function. A compact simulation study is carried out to examine the behavior of the proposed estimation methods. Finally, one real data analysis is performed for illustrative purposes.     

A literature review on planning and analysis of accelerated testing for reliability assessment

Accelerated testing has been widely used for several decades. Started with accelerated life tests with constant‐stress loadings, more interest has been focused prominently on accelerated degradation tests and time‐varying stress loadings. Because accelerated testing is crucial to the assessment of product reliability and the design of warranty policy, it is important to develop an efficacious test plan that encompasses and addresses important issues, such as design of stress profiles, sample allocation, test duration, measurement frequency, and budget constraint. In recent years, extensive research has been conducted on the optimal design of accelerated testing plans, and the consideration of multiple stresses with interactions has become a big challenge in such experimental designs. The purpose of this study is to provide a comprehensive review of important methods for statistical inference and optimal design of accelerated testing plans by compiling the existing body of knowledge in the area of accelerated testing. In this work, different types of test planning strategies are categorized, and their drawbacks and the research trends are provided to assist researchers and practitioners in conducting new research in this area.     

基于仿真的竞争失效产品综合应力加速寿命试验优化设计

针对竞争失效产品综合应力加速寿命试验存在试验时间长、费用高、效率低的问题,提出一种基于仿真的竞争失效产品综合应力加速寿命试验优化设计方法。采用Monte-Carlo进行仿真模拟,以正常使用应力下的p阶分位寿命渐进方差估计最小为目标,在各应力组合下试验截尾数大小给定的条件下,以试验应力水平大小作为设计变量,利用MLE理论进行统计分析,建立基于仿真的竞争失效产品综合应力加速寿命试验优化设计模型。通过实例分析表明该方法有效、可行,为电子装备寿命预测的加速试验方案优化设计提供技术支撑。     

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.     

Probabilistic cumulative damage model to estimate fatigue life

This paper introduces a numerical model to estimate fatigue life under step‐stress conditions, using the Weibull and lognormal distributions. The maximum likelihood method was used to estimate the free parameters of the distributions. The model was fitted to an experimental data on fatigue life in the specimens of steel SAE 8620, by using evolutionary computation to optimize the likelihood function. Results are reported on the values of the parameters and their confidence interval. Also, a validation of the model is discussed using analysis of residuals.     

Lower Percentile Estimation of Accelerated Life Tests with Nonconstant Scale Parameter

Lower percentiles of product lifetime are useful for engineers to understand product failure, and avoiding costly product failure claims. This paper proposes a percentile re‐parameterization model to help reliability engineers obtain a better lower percentile estimation of accelerated life tests under Weibull distribution. A log transformation is made with the Weibull distribution to a smallest extreme value distribution. The location parameter of the smallest extreme value distribution is re‐parameterized by a particular 100pth percentile, and the scale parameter is assumed to be nonconstant. Maximum likelihood estimates of the model parameters are derived. The confidence intervals of the percentiles are constructed based on the parametric and nonparametric bootstrap method. An illustrative example and a simulation study are presented to show the appropriateness of the method. The simulation results show that the re‐parameterization model performs better compared with the traditional model in the estimation of lower percentiles, in terms of Relative Bias and Relative Root Mean Squared Error. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

A residual stress dependent multiaxial fatigue life model of welded structures

In this paper, the influence of the residual stress on the fatigue performance of a welded structure under multiaxial loading modes is studied. First, the local stress state at weld toe is modified via introduction of the residual stress, and a new fatigue life estimation model considering the effect of the residual stress is established by modifying our recently proposed critical plane method. Second, the basic theory and procedure of the finite element simulation on the calculation of the welding residual stress are presented. Finally, a numerical simulation of an aluminum alloy flange‐to‐tube welding process is conducted, and the calculated residual stress is verified with X‐ray diffraction measurement. Furthermore, the performance of the proposed fatigue life estimation model is verified by the experimental data obtained in the fatigue test under different loading modes. It confirms that the consideration of the residual stress is important, especially under the out‐of‐phase loading mode.     

A comparison of two methods of analysing staircase fatigue test data

When the fatigue strength of components or material is tested by the staircase or up‐and‐down method, the data can be analysed in either the fatigue strength domain, which is the most popular approach, or the life domain, based on life regression models. In this paper we exploit a simple relationship between families of models in the two domains to obtain an estimate of the distribution of fatigue strength directly from a fitted life regression model. We use simulation to compare the two approaches, concluding that for estimating important features of the fatigue strength distribution the method of maximum likelihood is more effective if it is applied in the life domain. In particular, it is more robust to the choice of x_h, the step height of the staircase, provided x_h is in the range σ_X–2σ_X, where σ_X is the standard deviation of the log‐strengths. Most of our results are insensitive to the value of the initial stress amplitude. Copyright © 2007 John Wiley & Sons, Ltd.