A general Bayes weibull inference model for accelerated life testing

This article presents the development of a general Bayes inference model for accelerated life testing. The failure times at a constant stress level are assumed to belong to a Weibull distribution, but the specification of strict adherence to a parametric time-transformation function is not required. Rather, prior information is used to indirectly define a multivariate prior distribution for the scale parameters at the various stress levels and the common shape parameter. Using the approach, Bayes point estimates as well as probability statements for use-stress (and accelerated) life parameters may be inferred from a host of testing scenarios. The inference procedure accommodates both the interval data sampling strategy and type I censored sampling strategy for the collection of ALT test data. The inference procedure uses the well-known MCMC (Markov Chain Monte Carlo) methods to derive posterior approximations. The approach is illustrated with an example.     

Discrimination between accelerated life models via Approximate Bayesian Computation

Accelerated life testing (ALT) is widely used in high-reliability product estimation to get relevant information about an item's performance and its failure mechanisms. To analyse the observed ALT data, reliability practitioners need to select a suitable accelerated life model based on the nature of the stress and the physics involved. A statistical model consists of (i) a lifetime distribution that represents the scatter in product life and (ii) a relationship between life and stress. In practice, several accelerated life models could be used for the same failure mode and the choice of the best model is far from trivial. For this reason, an efficient selection procedure to discriminate between a set of competing accelerated life models is of great importance for practitioners. In this paper, accelerated life model selection is approached by using the Approximate Bayesian Computation (ABC) method and a likelihood-based approach for comparison purposes. To demonstrate the efficiency of the ABC method in calibrating and selecting accelerated life model, an extensive Monte Carlo simulation study is carried out using different distances to measure the discrepancy between the empirical and simulated times of failure data. Then, the ABC algorithm is applied to real accelerated fatigue life data in order to select the most likely model among five plausible models. It has been demonstrated that the ABC method outperforms the likelihood-based approach in terms of reliability predictions mainly at lower percentiles particularly useful in reliability engineering and risk assessment applications. Moreover, it has shown that ABC could mitigate the effects of model misspecification through an appropriate choice of the distance function.     

Bayes approach in RDT using accelerated and long-term life data

A common problem of reliability demonstration testing (RDT) is the magnitude of total time on test required to demonstrate reliability to the consumer’s satisfaction, particularly in the case of high reliability components. One solution is the use of accelerated life testing (ALT) techniques. Another is to incorporate prior beliefs, engineering experience, or previous data into the testing framework. This may have the effect of reducing the amount of testing required in the RDT in order to reach a decision regarding conformance to the reliability specification. It is in this spirit that the use of a Bayesian approach can, in many cases, significantly reduce the amount of testing required.We demonstrate the use of this approach to estimate the acceleration factor in the Arrhenius reliability model based on long-term data given by a manufacturer of electronic components (EC). Using the Bayes approach we consider failure rate and acceleration factor to vary randomly according to some prior distributions. Bayes approach enables for a given type of technology the optimal choice of test plan for RDT under accelerated conditions when exacting reliability requirements must be met. These requirements are given by a hypothetical consumer by two different ways. The calculation of posterior consumer’s risk is demonstrated in both cases.The test plans are optimum in that they take into account Var{λ|data}, posterior risk, E{λ|data}, Median λ or other percentiles of λ at data observed at the accelerated conditions. The test setup assumes testing of units with time censoring.     

Accelerated Reliability Demonstration Testing Design Based on Reliability Allocation of Environmental Stresses

The traditional reliability demonstration testing based on statistical method requires a large number of samples and long testing time, failing to satisfy the demand for short cycle and low cost. This paper proposes a new accelerated approach for determining reliability target of each environment stress, accelerated test profile, and comprehensive acceleration factor for multi‐failure mode product to conduct assembly level accelerated demonstration testing under multiple stresses and levels. By decomposing the product from four levels, namely function, structure, mechanism, and stress, the products' weaknesses can be identified. The main failure modes and sensitive environmental stresses are determined based on environmental profile and FMMEA. In this design, the reliability is apportioned to each actual environmental stress by AHP. And the SSI model is used to establish accelerated stress profile. The overall acceleration factor can be derived from the model of assembly level accelerated testing. Combined with a statistical plan, the accelerated reliability demonstration testing plan and test profile is built. A case example is presented to illustrate the effectiveness of the proposed approach. Copyright © 2017 John Wiley & Sons, Ltd.     

Using Accelerated Life Tests Results to Predict Product Field Reliability

Accelerated life tests (ALTs) provide timely assessments of the reliability of materials, components, and subsystems. ALTs can be run at any of these levels or at the full-system level. Sometimes ALTs generate multiple failure modes. A frequently asked question near the end of an ALT program is “what do these test results say about field performance?” ALTs are carefully controlled, whereas the field environment is highly variable. For example, products in the field have different average use rates across the product population. With good characterization of field use conditions, it may be possible to use ALT results to predict the failure time distribution in the field. When such information is not available but both life test data and field data (from, e.g., warranty returns) are available, it may be possible to find a model to relate the two data sets. Under a reasonable set of practical assumptions, this model then can be used to predict the failure time distribution for a future component or product operating in the same use environment. This paper describes a model and methods for such situations. The methods are illustrated by an example to predict the failure time distribution of a newly designed product with two failure modes. Supplemental material for this article is available online at the Technometrics website.     

Index to Volume 32

ABSTRACT

Most of the recently developed methods on optimum planning for accelerated life tests (ALT) involve “guessing” values of parameters to be estimated, and substituting such guesses in the proposed solution to obtain the final testing plan. In reality, such guesses may be very different from true values of the parameters, leading to inefficient test plans. To address this problem, we propose a sequential Bayesian strategy for planning of ALTs and a Bayesian estimation procedure for updating the parameter estimates sequentially. The proposed approach is motivated by ALT for polymer composite materials, but are generally applicable to a wide range of testing scenarios. Through the proposed sequential Bayesian design, one can efficiently collect data and then make predictions for the field performance. We use extensive simulations to evaluate the properties of the proposed sequential test planning strategy. We compare the proposed method to various traditional non-sequential optimum designs. Our results show that the proposed strategy is more robust and efficient, as compared to existing non-sequential optimum designs. Supplementary materials for this article are available online.     

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.     

Reliability analysis of exponentiated Poisson-exponential constant stress accelerated life test model

Accelerated life testing is an efficient tool frequently adopted for obtaining failure time data of test units in a lesser time period as compared to normal use conditions. We assume that the lifetime data of a product at constant level of stress follows an exponentiated Poisson-exponential distribution and the shape parameter of the model has a log-linear relationship with the stress level. Model parameters, the reliability function (RF), and the mean time to failure (MTTF) function under use conditions are estimated based on eight frequentist methods of estimation, namely, method of maximum likelihood, method of least square and weighted least square, method of maximum product of spacing, method of minimum spacing absolute-log distance, method of Cramér-von-Mises, method of Anderson–Darling, and Right-tail Anderson–Darling. The performance of the different estimation methods is evaluated in terms of their mean relative estimate and mean squared error using small and large sample sizes through a Monte Carlo simulation study. Finally, two accelerated life test data sets are considered and bootstrap confidence intervals for the unknown parameters, predicted shape parameter, predicted RF, and the MTTF at different stress levels, are obtained.     

Generation and validation of loading profiles for highly accelerated durability tests of ground vehicle components

Accelerated durability tests are designed to quantify the life characteristics of ground vehicle components under normal use conditions by testing at a higher stress level to accelerate the occurrence of failure. Presently, conducting durability tests with a high acceleration factor has become increasingly demanding for the reduction of the time and the cost involved in long period field/durability tests. In previous work, to accelerate the field test, the standard ‘test tailoring approach’ has been modified due to the limitations of testing implementation and required high acceleration factors. In this modified approach, a full period durability loading profile has to be shortened to an equivalent partial period test loading profile, which is repeated in the tests keeping the same amount of damage contents. To apply this new modified approach to industrial durability tests, it needs to be validated. In this work, a computer-aided testing method is developed for the validation of this modified ‘test tailoring approach’. Hence, a new test-piece has been designed by a conjugative approach involving the finite element technique and fatigue analysis for a specific durability life. Afterwards, the loading profiles with various acceleration factors synthesized via the modified approach have been applied on the designed test-piece and the fatigue lives have been simulated to verify the effectiveness of those loading profiles. Simulation results show that, loading profiles with high acceleration factors can be successfully generated with the accuracy above 95%. In addition, synthesized accelerated loading profiles result failure from the identical locations determined using the proposed conjugative approach.     

Integration of computation and testing for reliability estimation

This paper develops a methodology to integrate reliability testing and computational reliability analysis for product development. The presence of information uncertainty such as statistical uncertainty and modeling error is incorporated. The integration of testing and computation leads to a more cost-efficient estimation of failure probability and life distribution than the tests-only approach currently followed by the industry. A Bayesian procedure is proposed to quantify the modeling uncertainty using random parameters, including the uncertainty in mechanical and statistical model selection and the uncertainty in distribution parameters. An adaptive method is developed to determine the number of tests needed to achieve a desired confidence level in the reliability estimates, by combining prior computational prediction and test data. Two kinds of tests — failure probability estimation and life estimation — are considered. The prior distribution and confidence interval of failure probability in both cases are estimated using computational reliability methods, and are updated using the results of tests performed during the product development phase.     

融合纤维数据的复合材料筒体结构持久寿命评估方法

对于由T700碳纤维浸胶丝束缠绕铝合金筒壁构成的复合材料筒体结构, 通过提高筒体转速增加纤维拉应力的加速寿命试验仍无法获得充分的失效数据, 试验结果多表现为高应力大删失比与低应力无失效混合的寿命数据类型, 且采用现行方法进行持久寿命评估的结果存在置信度偏低的问题。针对这一问题, 首先对纤维和筒体的加速等同性与失效机制一致性进行分析, 进而通过对加速模型参数估计量的部分协方差矩阵进行统计检验与信息融合, 建立了一种可融合纤维数据的筒体结构持久寿命评估方法。该方法通过综合利用碳纤维的相似性信息, 从而提高筒体结构持久寿命评估的精度。采用实例验证了文中方法的有效性, 使筒体结构持久寿命的评估精度提高了35%。     

Optimal accelerated life tests under a cost constraint with non-uniform stress durations

ABSTRACT

To collect the information about the lifetime distribution of a product, a standard life testing method at normal working conditions is impractical when the product has a substantially long lifespan. Accelerated life testing solves this problem by subjecting the test units at higher stress levels for quicker and more failure data. Due to constrained resources in practice, several decision variables such as the allocation proportions and stress durations must be determined carefully at the design stage in order to run an accelerated life test efficiently. These decision variables directly affect the experimental cost as well as the estimate precision of the parameters of interest. This article investigates these optimal decision variables based on several well-known optimality criteria under the constraint that the total experimental cost does not exceed a pre-specified budget. A general scale family of distributions is considered for the underlying lifetimes to accommodate different lifetime models at different stress levels for flexible modeling. The constant-stress and step-stress accelerated life tests are then studied in detail with linearly decreasing stress durations as the stress level progresses. Under the identical budget constraint, the efficiencies of these two stress loading schemes are compared using two case studies.     

Reliability Estimation for Accelerated Life Tests Based on a Cox Proportional Hazard Model with Error Effect

In this paper, a Cox proportional hazard model with error effect applied on the study of an accelerated life test is investigated. Statistical inference under Bayesian methods by using the Markov chain Monte Carlo techniques is performed in order to estimate the parameters involved in the model and predict reliability in an accelerated life testing. The proposed model is applied to the analysis of the knock sensor failure time data in which some observations in the data are censored. The failure times at a constant stress level are assumed to be from a Weibull distribution. The analysis of the failure time data from an accelerated life test is used for the posterior estimation of parameters and prediction of the reliability function as well as the comparisons with the classical results from the maximum likelihood estimation. Copyright © 2017 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.     

Utilizing Accelerated Degradation and Field Data for Life Prediction of Highly Reliable Products

For newly developed, highly reliable, and long‐lifespan products, it is quite difficult to implement effective remaining useful life (RUL) prediction in the early usage under limited time cost. However, accelerated degradation testing (ADT) is generally used for lifetime evaluation for such products with harsher test conditions and shorter test time in the late research and development phase. Thus, in this paper, we propose a life prediction framework to integrate the information from ADT to conduct field RUL prediction for highly reliable products. Because ADT belongs to reliability testing used for inferring the population information from the selected test samples, we at first present the modified Wiener process (MWP) model. Different from traditional methods that embody both the random variability and unit‐to‐unit variability into the diffusion coefficient, the proposed method describes them separately in ADT analysis. Then, the MWP model from ADT is used as a prior for field RUL prediction of the target product during which the strong tracking filtering algorithm is introduced for updating the hidden state and computing the RUL prediction results when the new monitoring data are available. Because of the complexity of the MWP model, the Markov chain Monte Carlo method is provided to estimate the unknown parameters. Finally, the simulation study and the light‐emitting diode application verify the effectiveness of the proposed framework that can achieve reasonable life prediction results for highly reliable products for both linear and nonlinear scenarios. Copyright © 2015 John Wiley & Sons, Ltd.     

Integrated Approach for Field Reliability Prediction Based on Accelerated Life Testing

ABSTRACT

To predict field reliability using analytical modeling, several important reliability activities should be conducted, including failure mode and effect analysis, stress and usage condition analysis, physics of failure analysis, accelerated life testing and modeling, and cumulative damage modeling if needed. With all of the mentioned activities and results, the field reliability confidence limit can be predicted at a certain confidence level, if a modeling framework can be established. This article builds such an integrated process and comprehensive modeling framework, especially with cumulative damage rules when the certain field stresses are random processes. An engineering product is provided as an application to illustrate the effectiveness of proposed method.     

Field Failure Data: an Alternative Proposal for Reliability Estimation

Product reliability is a very important issue for the competitive strategy of industries. In order to estimate a product's reliability, parametric inferential methods are required to evaluate survival test data, which happens to be a fairly expensive data source. Such costly information usually imposes additional compromises in the product development and new challenges to be overcome throughout the product's life cycle. However, manufacturers also keep field failure data for warranty and maintenance purposes, which can be a low‐cost data source for reliability estimation. Field‐failure data are very difficult to evaluate using parametric inferential methods due to their small and highly censored samples, quite often representing mixed modes of failure. In this paper a method for reliability estimation using field failure data is proposed. The proposal is based on the use of non‐parametric inferential methods, associated with resampling techniques to derive confidence intervals for the reliability estimates. Test results show the adequacy of the proposed method to calculate reliability estimates and their confidence interval for different populations, including cases with highly right‐censored failure data. The method is shown to be particularly useful when the sampling distribution is not known, which happens to be the case in a large number of practical reliability evaluations. Copyright © 2006 John Wiley & Sons, Ltd.     

基于Bayes法的电主轴极小子样可靠性研究

针对长寿命的磨削电主轴极小子样的可靠性评估问题,提出了Bayes结合虚拟增广样本的分析方法。首先,在Bayes法基本流程的指导下,研究了基于Bayes法的磨削电主轴可靠性评估方法。根据定时截尾试验的原则对电主轴进行可靠性试验,应用Bayes法结合磨削电主轴试验样本的可靠性试验数据,综合虚拟增广样本法对其可靠性进行评估,最终获得电主轴的可靠性评估结果。最后,将基于Bayes法与基于伪寿命分布法的磨削电主轴极小子样可靠性评估结果进行比较,以验证基于Bayes法可靠性评估理论的合理性。     

Reliability estimation by ALT when no analytical model holds

This paper presents an accelerated life testing method applicable to devices or systems when no analytical relationship with respect to the stress level can be defined. If a numerical approach remains possible, the numerical model can be fitted to the accelerated test results. Thus, long‐term failures can be predicted from short tests. This method is carried out in the case of fatigue, the evolution of the damage leading to the failure having to be modeled by a numerical finite element method. Copyright © 2010 John Wiley & Sons, Ltd.