Optimum simple step-stress accelerated life tests for weibull distribution and type I censoring

This article presents an optimum simple step-stress accelerated life test for the Weibull distribution under Type I censoring. It is assumed that a log-linear relationship exists between the Weibull scale parameter and the (possibly transformed) stress and that a certain cumulative exposure model for the effect of changing stress holds. The optimum plan—low stress and stress change time—is obtained, which minimizes the asymptotic variance of the maximum likelihood estimator of a stated percentile at design stress. For selected values of the design parameters, nomographs useful for finding the optimum plan are given, and the effects of errors in preestimates of the parameters are investigated. As an alternative to the simple step-stress test, a three-level compromise plan is proposed, and its performance is studied and compared with that of the optimum simple step-stress test. © 1993 John Wiley & Sons. Inc.     

Optimum step‐stress accelerated life test plans for log‐location‐scale distributions

This article presents new tools and methods for finding optimum step‐stress accelerated life test plans. First, we present an approach to calculate the large‐sample approximate variance of the maximum likelihood estimator of a quantile of the failure time distribution at use conditions from a step‐stress accelerated life test. The approach allows for multistep stress changes and censoring for general log‐location‐scale distributions based on a cumulative exposure model. As an application of this approach, the optimum variance is studied as a function of shape parameter for both Weibull and lognormal distributions. Graphical comparisons among test plans using step‐up, step‐down, and constant‐stress patterns are also presented. The results show that depending on the values of the model parameters and quantile of interest, each of the three test plans can be preferable in terms of optimum variance. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Optimal design of accelerated life tests under periodic inspection

For the exponentially distributed lifetimes, optimal accelerated life test plans are determined under the assumptions of periodic inspection and Type I censoring. Computational results indicate that for the range of parameter values considered the asymptotic variance of the estimated mean or pth quantile at the use stress is not sensitive to the number of inspections at overstress levels. Senstivity analyses are also conducted to see how senstive the asymptotic variance of the estimated mean is with respect to the uncertainties involved in the guessed failure probabilities at the use and high stress levels. Computational results show that moderate deviations (several tens of percents) of the guessed failure probabilities from their true values are fairly tolerable in terms of the relative amount of increase in the asymptotic variance of the estimated mean. Procedures for selecting a sample size and for determining whether or not to conduct an accelerated life test are also discussed.     

定数截尾的恒定应力加速寿命试验的优化设计

在指数分布场合下讨论了恒加试验的最优设计问题 ,得到了一系列与步加试验相对应的结果 .这里的最优准则是指正常应力水平下产品对数平均寿命估计的方差的极小化 .     

Designing an optimal life test with type I censoring

The present study considers the design of an optimal life test plan with Type I censoring in the case where only one testing machine is available and a machine tests items sequentially as seen in fatigue life tests of metallic materials. Under the condition that the life test is conducted in a fixed censoring time and with a fixed number of test items, a method for determining the censoring time and the number of test items is proposed from the Bayesian point of view. The proposed method is also applied to the Weibull life distribution with a known shape parameter.     

Accelerated life test plans under intermittent inspection and type-I censoring: The case of weibull failure distribution

For the case where the lifetime at a stress level has a Weibull distribution, statistically optimal and practical accelerated life test plans are developed under the assumptions of intermittent inspection and Type-I censoring. For a statistically optimal plan the low stress level, the proportion of test units allocated, and the inspection times are determined such that the asymptotic variance of the maximum-likelihood estimator of a certain quantile at the use condition is minimized. Although the practical plan adopts the same design criterion, it involves three rather than two overstress levels and easily calculated inspection schemes. Despite some loss in efficiency the practical plan has several advantages over the statistically optimal one. For instance, the practical plan can provide means for checking the validity of the assumptions made, may reduce the danger of extrapolation, and is more convenient to determine and implement. Computational experiments are conducted to evaluate the relative efficiency of a practical plan to the corresponding statistically optimal plan. Guidelines for selecting an appropriate practical plan are also described with an example.     

Planning accelerated life tests for censored two‐parameter exponential distributions

We consider optimal test plans involving life distributions with failure‐free life, i.e., where there is an unknown threshold parameter below which no failure will occur. These distributions do not satisfy the regularity conditions and thus the usual approach of using the Fisher information matrix to obtain an optimal accelerated life testing (ALT) plan cannot be applied. In this paper, we assume that lifetime follows a two‐parameter exponential distribution and the stress‐life relationship is given by the inverse power law model. Near‐optimal test plans for constant‐stress ALT under both failure‐censoring and time‐censoring are obtained. We first obtain unbiased estimates for the parameters and give the approximate variance of these estimates for both failure‐censored and time‐censored data. Using these results, the variance for the approximate unbiased estimate of a percentile at a design stress is computed and then minimized to produce the near‐optimal plan. Finally, a numerical example is presented together with simulation results to study the accuracy of the approximate variance given by the proposed plan and show that it outperforms the equal‐allocation plan. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 169–186, 1999     

A nonparametric approach to accelerated life testing under multiple stresses

Accelerated life testing (ALT) is concerned with subjecting items to a series of stresses at several levels higher than those experienced under normal conditions so as to obtain the lifetime distribution of items under normal levels. A parametric approach to this problem requires two assumptions. First, the lifetime of an item is assumed to have the same distribution under all stress levels, that is, a change of stress level does not change the shape of the life distribution but changes only its scale. Second, a functional relationship is assumed between the parameters of the life distribution and the accelerating stresses. A nonparametric approach, on the other hand, assumes a functional relationship between the life distribution functions at the accelerated and nonaccelerated stress levels without making any assumptions on the forms of the distribution functions. In this paper, we treat the problem nonparametrically. In particular, we extend the methods of Shaked, Zimmer, and Ball [7] and Strelec and Viertl [8] and develop a nonparametric estimation procedure for a version of the generalized Arrhenius model with two stress variables assuming a linear acceleration function. We obtain consistent estimates as well as confidence intervals of the parameters of the life distribution under normal stress level and compare our nonparametric method with parametric methods assuming exponential, Weibull and lognormal life distributions using both real life and simulated data. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 629–644, 1998     

Optimal design of accelerated life tests with two stresses

Most of the previous works on designing accelerated life test plans were concerned with the case where a single stress is employed for acceleration. In this article we develop optimal accelerated life test plans when two stresses are involved with possible interaction between them. The lifetimes of test items are assumed to follow an exponential distribution, the mean of which depends on the stresses according to the generalized Eyring law. A factorial arrangement of test points is considered for an efficient utilization of equipment, and the low level of each stress and the proportion of test items allocated to each test point are determined such that the asymptotic variance of the maximum-likelihood estimator of the mean lifetime at the use condition or of an acceleration factor is minimized. Patterns of optimal plans are identified and their efficiencies are compared with the corresponding single-stress accelerated life test plans. © 1996 John Wiley & Sons, Inc.     

Comparison between constant‐stress and step‐stress accelerated life tests under Time Constraint

By running life tests at higher stress levels than normal operating conditions, accelerated life testing (ALT) quickly yields information on the lifetime distribution of a test unit. The lifetime at the design stress is then estimated through extrapolation using a regression model. In constant‐stress testing, a unit is tested at a fixed stress level until failure or the termination time point of test, whereas step‐stress testing allows the experimenter to gradually increase the stress levels at some prefixed time points during the test. In this work, the optimal k‐level constant‐stress and step‐stress ALTs are compared for the exponential failure data under complete sampling and Type‐I censoring. The objective is to quantify the advantage of using the step‐stress testing relative to the constant‐stress one. Assuming a log‐linear life–stress relationship with the cumulative exposure model for the effect of changing stress in step‐stress testing, the optimal design points are determined under C/D/A‐optimality criteria. The efficiency of step‐stress testing to constant‐stress one is then discussed in terms of the ratio of optimal objective functions based on the information matrix. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 00: 000–000, 2013     

Priori information analysis of optocoupler accelerated degradation experiment based on failure mechanism verification test

The optocoupler is a weak link in the inertial navigation platform of a kind of guided munitions. It is necessary to use accelerated storage test to verify the storage life of long storage products. Especially for small sample products, it is very important to obtain prior information for the design and implementation of accelerated degradation test. In this paper, the optocoupler failure mechanism verification test is designed and the experimental results are analyzed and the prior information is obtained. The results show that optocouplers have two failure modes, one is sudden failure and the other is degradation failure; the maximum temperature stress of optocoupler can't exceed 140 °C; the increase of leakage current of optocoupler is caused by movable ions contaminating the LED chip. The surface leakage current is proportional to the adsorption amount. The increase of leakage current makes p-n junction tunneling effect occur which LEDs the failure of the optocoupler. The lifetime distribution model of the optocoupler is determined by the failure physics. The lifetime of the optocoupler is subject to the lognormal distribution. The degeneracy orbit of the optocoupler leakage current is described by a power law model. The estimated values of the orbital parameters are initially calculated and the parameters of its life distribution function are deduced. The above information lays a good foundation for the optimization design and data processing of the accelerated degradation experiment.     

加速寿命试验下复合失效产品的可靠性评估

产品往往受随机失效和耗损失效两种模式的双重影响,通过构建复合寿命分布模型来表征这类产品失效规律。在加速寿命试验下,基于极大似然估计构造了该类产品寿命参数估计的数值算法;为了验证该寿命分布是否真服从假定的复合分布,构建了常应力等效失效数据的柯尔莫哥诺夫分布检验法;最后,以电连接器这一典型产品为例进行了工程仿真试验并进行数据评估,从而验证了评估方法的正确性。     

Stage life testing

In progressive censoring, items are removed at certain times during the life test. Commonly, it is assumed that the removed items are used for further testing. In order to take into account information about these additional testing in inferential procedures, we propose a two‐step model of stage life testing with one fixed stage‐change time which incorporates information about both the removed items (further tested under different conditions) and those remaining in the current life test. We show that some marginal distributions in our model correspond either to progressive censoring with a fixed censoring time or to a simple‐step stress model. Furthermore, assuming a cumulative exposure model, we establish exact inferential results for the distribution parameters when the lifetimes are exponentially distributed. An extension to Weibull distributed lifetimes is also discussed.     

A model for step‐stress accelerated life testing

Modern technology is producing high reliability products. Life testing for such products under normal use condition takes a lot of time to obtain a reasonable number of failures. In this situation a step‐stress procedure is preferred for accelerated life testing. In this paper we assume a Weibull and Lognormal model whose scale parameter depends upon the present level as well as the age at the entry in the present stress level. On the basis of that we propose a parametric model to the life distribution for step‐stress testing and suggest a suitable design to estimate the parameters involved in the model. A simulation study has been done by the proposed model based on maximum likelihood estimation. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

一种新的Weibull分布恒定应力加速寿命试验分析方法

分析了Weibull分布恒定应力加速寿命试验常用的二步分析方法存在的不足 ,建立了一种新的构造数据分析方法。考虑到分布参数的相关性 ,该方法引入了Weibull分布特征寿命参数的二次估计 ,在模型拟合优度上高于原来的二步分析方法 ,使分析精度得到了改善。同时 ,该方法避免了原来二步分析方法的查表过程 ,便于软件实现和工程实际应用。     

A general model for accelerated life testing with time‐dependent covariates

This paper introduces a general or “distribution‐free” model to analyze the lifetime of components under accelerated life testing. Unlike the accelerated failure time (AFT) models, the proposed model shares the advantage of being “distribution‐free” with the proportional hazard (PH) model and overcomes the deficiency of the PH model not allowing survival curves corresponding to different values of a covariate to cross. In this research, we extend and modify the extended hazard regression (EHR) model using the partial likelihood function to analyze failure data with time‐dependent covariates. The new model can be easily adopted to create an accelerated life testing model with different types of stress loading. For example, stress loading in accelerated life testing can be a step function, cyclic, or linear function with time. These types of stress loadings reduce the testing time and increase the number of failures of components under test. The proposed EHR model with time‐dependent covariates which incorporates multiple stress loadings requires further verification. Therefore, we conduct an accelerated life test in the laboratory by subjecting components to time‐dependent stresses, and we compare the reliability estimation based on the developed model with that obtained from experimental results. The combination of the theoretical development of the accelerated life testing model verified by laboratory experiments offers a unique perspective to reliability model building and verification. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 303–321, 1999     

Progressive Type‐II censoring and coherent systems

A new connection between the distribution of component failure times of a coherent system and (adaptive) progressively Type‐II censored order statistics is established. Utilizing this property, we develop inferential procedures when the data is given by all component failures until system failure in two scenarios: In the case of complete information, we assume that the failed component is also observed whereas in the case of incomplete information, we have only information about the failure times but not about the components which have failed. In the first setting, we show that inferential methods for adaptive progressively Type‐II censored data can directly be applied to the problem. For incomplete information, we face the problem that the corresponding censoring plan is not observed and that the available inferential procedures depend on the knowledge of the used censoring plan. To get estimates for distributional parameters, we propose maximum likelihood estimators which can be obtained by solving the likelihood equations directly or via an Expectation‐Maximization‐algorithm type procedure. For an exponential distribution, we discuss also a linear estimator to estimate the mean. Moreover, we establish exact distributions for some estimators in the exponential case which can be used, for example, to construct exact confidence intervals. The results are illustrated by a five component bridge system. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 512–530, 2015     

Bayesian estimation and optimal designs in partially accelerated life testing

A method of life testing is proposed which combines both ordinary and accelerated life-testing procedures. It is assumed that an item can be tested either in a standard environment or under stress. The amount of stress is fixed in advance and is the same for all items to be tested However, the time x at which an item on lest is taken out of the standard environment and put under stress can be chosen by the experimenter subject to a given cost structure. When an item is put under stress its lifetime is changed by the factor α. Let the random variable T denote the lifetime of an item in the standard environment, and let γ denote its lifetime under the partially accelerated test procedure just described. Then Y = T if T ≦ x, and Y = x + α (T > x) if T > x. It is assumed that T has an exponential distribution with parameter θ. The estimation of θ and α and the optimal design of a partially accelerated life test are studied in the framework of Bayesian decision theory.     

Designing acceptance sampling schemes for life testing with mixed censoring

Mixed censoring is useful extension of Type I and Type II censoring and combines some advantages of both types of censoring. This paper proposes a general Bayesian framework for designing a variable acceptance sampling scheme with mixed censoring. A general loss function which includes the sampling cost, the time‐consuming cost, the salvage value, and the decision loss is employed to determine the Bayes risk and the corresponding optimal sampling plan. An explicit expression of the Bayes risk is derived. The new model can easily be adapted to create life testing models for different distributions. Specifically, two commonly used distributions including the exponential distribution and the Weibull distribution are considered with a special decision loss function. We demonstrate that the proposed model is superior to models with Type I or Type II censoring. Numerical examples are reported to illustrate the effectiveness of the method proposed. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

MAD: Mathematical analysis of downtime

The MAD model presents a mathematic treatment of the relationship between aircraft reliability and maintainability, system manning and inspection policies, scheduling and sortie length, and aircraft downtime. Log normal distributions are postulated for subsystem repair times and simultaneous repair of malfunctions is assumed. The aircraft downtime for maintenance is computed with the distribution of the largest of k log normal distributions. Waiting time for maintenance men is calculated either by using a multiple-channel queuing model or by generating the distribution of the number of maintenance men required and comparing this to the number of men available to determine the probability of waiting at each inspection.