Step-stress life-testing with random stress-change times forexponential data

This paper studies statistical models in step-stress accelerated life-testing when the stress-change times are random. The marginal lifetime distribution of a test unit under a step-stress test plan when the stress change times are random variables is presented. Maximum likelihood estimates for model parameters based on both the marginal and conditional life distributions are considered. An optimum test plan is explored for simple step-stress test when the stress change time is an order statistic from the exponential lifetime under the low-stress level     

Analysis of grouped and censored data from step-stress life test

This paper studies statistical analysis of grouped and censored data obtained from a step-stress accelerated life test. We assume that the stress change times in the step-stress life test are fixed and the lifetimes observed are type I censored. Maximum likelihood estimates and asymptotic confidence intervals for model parameters are obtained. We provide an asymptotic statistical test for the cumulative exposure model based on the grouped and type I censored data. We also present the optimum test plan for a simple step-stress test when the lifetime under constant stress is assumed exponential. Finally we give an application of our methods by applying our analysis process to a real life data set. The proposed statistical methodology is especially useful when intermittent inspection is the only feasible way of checking the status of test units during a step-stress test.     

Optimal simple step-stress plan for cumulative exposure model using log-normal distribution

Optimal times of changing stress level for simple step-stress plans under a cumulative exposure model using the log-normal distribution are determined for a wide range of values of the parameters in the model. A table of optimal times of changing stress level for various model parameters values is obtained. A formula for optimal time of changing stress level is also estimated from the table. This paper provides an optimal life testing plan which will enable us to accurately estimate the 50th percentile of the life time of a product being tested without having to wait long time for the product to fail.     

Accelerated Life Testing?Step-Stress Models and Data Analyses

This paper presents statistical models and methods for analyzing accelerated life-test data from step-stress tests. Maximum likelihood methods provide estimates of the parameters of such models, the life distribution under constant stress, and other information. While the methods are applied to the Weibull distribution and inverse power law, they apply to many other accelerated life test models. These methods are illustrated with step-stress data on time to breakdown of an electrical insulation.     

Optimum Simple Step-Stress Plans for Accelerated Life Testing

This paper presents optimum plans for simple (two stresses) step-stress tests where all units are run to failure. Such plans minimize the asymptotic variance of the maximum likelihood estimator (MLE) of the mean life at a design stress. The life-test model consists of: 1) an exponential life distribution with 2) a mean that is a log-linear function of stress, and 3) a cumulative exposure model for the effect of changing stress. Two types of simple step-stress tests are considered: 1) a time-step test and 2) a failure-step test. A time-step test runs a specified time at the first stress, whereas, a failure-step test runs until a specified proportion of units fail at the first stress. New results include: 1) the optimum time at the first stress for time-step test and 2) the optimum proportion failing at the low stress for a failure-step test, and 3) the asymptotic variance of these optimum tests. Both the optimum time-step and failure-step tests have the same asymptotic variance as the corresponding optimum constant-stress test. Thus step-stress tests yield the same amount of information as constant-stress tests.     

Some properties of confidence bounds on reliability estimation for parts under varying stresses

Reliability estimation is usually performed on a part under a constant stress level. However, a part could experience several different stress levels, or profiled stress, during its lifetime. One such example is when the part is subject to step-stress accelerated life testing. Studying the reliability estimation & its confidence bounds for a part under varying stresses will generalize the existing estimation methods for accelerated life testing. In this paper, we derive the reliability function of a part under varying stresses based on a Weibull failure time distribution, and cumulative damage model. The reliability confidence bounds, based on a s-normal approximation, are given explicitly, and their limiting properties are discussed. A step-stress accelerated life testing example is used to illustrate these interesting properties, which provides the insights of the limitation of the current test plan, and how to design a better one.     

Inferences on a simple step-stress model with type-II censoredexponential data

This paper presents the inferences of parameters on the simple step-stress model in accelerated life testing with type-II censoring. An exponential life distribution with a mean that is a log-linear function of stress, and a cumulative exposure model are assumed. Maximum likelihood methods are used to estimate the parameters of these models. Hypothesis tests about model parameters are discussed. Confidence intervals for parameter-estimates are constructed using a pivotal quantity. The advantage of these results is that they apply to any sample size. A numerical example illustrates the inferential procedure     

Weibull分布下基于MLE的红外发光二极管寿命预测

为了对红外发光二极管(LED)恒定及步进应力加速寿命试验的数据进行统计分析,应用Weibull分布函数描述了其寿命分布,利用极大似然法(MLE)及其迭代流程图估计出形状参数和尺度参数,通过最小二乘法确定了红外LED加速寿命方程,对红外LED寿命是否符合威布尔分布进行了Kolmogorov-Smirnov检验,并利用自行开发的寿命预测软件计算出平均寿命和中位寿命。数值结果表明,红外LED的寿命服从Weibull分布,加速寿命方程符合逆幂定律,所估计出的红外LED的寿命对生产厂商和用户有很强的指导意义。     

A least-squares approach to analyzing life-stress relationship instep-stress accelerated life tests

A new approach is proposed for the data analysis of failure-censored step-stress accelerated life tests with exponential failures, following an assumed log-linear life-stress relationship. This approach, transformed least squares: (1) provides closed-form parameter estimates; (2) motivates a simple statistic for life-stress relationship verification; and (3) appreciably simplifies the derivation of optimal test plans. The proposed D-statistic has good power and compares reasonably well with likelihood-ratio test statistic. Estimates are unbiased and exact for failure-step tests of any sample size. The derived optimal test plans agree with those derived using maximum likelihood estimation     

一种步进应力加速寿命试验数理统计分析法

本文提出了处理步进应力加速寿命试验的“时间等效”数学模型,并用它把寿命分布服从对数正态分布(或威布尔分布)、有m步的一次步进应力加速寿命试验变为有m级应力的恒定应力加速寿命试验来分析处理.并且推导出了它们之间的变换计算公式.用本法处理质子轰击隔离WC56型砷化镓场效应管步进应力加速寿命试验,获得了满意的结果.     

Inference for a Simple Step-Stress Model With Type-II Censoring, and Weibull Distributed Lifetimes

The simple step-stress model under Type-II censoring based on Weibull lifetimes, which provides a more flexible model than the exponential model, is considered in this paper. For this model, the maximum likelihood estimates (MLE) of its parameters, as well as the corresponding observed Fisher Information Matrix, are derived. The likelihood equations do not lead to closed-form expressions for the MLE, and they need to be solved by using an iterative procedure, such as the Newton-Raphson method. We also present a simplified estimator, which is easier to compute, and hence is suitable to use as an initial estimate in the iterative process for the determination of the MLE. We then evaluate the bias, and mean square error of these estimates; and provide asymptotic, and bootstrap confidence intervals for the parameters of the Weibull simple step-stress model. Finally, the results are illustrated with some examples.      

Testing exponentiality based on the Kullback-Leibler information with the type II censored data

We express the joint entropy of order statistics in terms of an incomplete integral of the hazard function, and provide a simple estimate of the joint entropy of the type II censored data. Then we establish a goodness of fit test statistic based on the Kullback-Leibler information with the type II censored data, and compare its performance with some leading test statistics. A Monte Carlo simulation study shows that the proposed test statistic shows better powers than some leading test statistics against the alternatives with monotone increasing hazard functions.     

Order Restricted Inference for Exponential Step-Stress Models

In the context of multiple step-stress models, which is a special type of accelerated life-testing model, interest lies on the expected lifetimes of the experimental units under different stress levels. Although the expected lifetime is shortened as the stress level increases, this information has not been incorporated so far into the associated inferential procedures. For this reason, we develop here the order restricted maximum likelihood estimation (MLE) for multiple step-stress models with exponentially distributed lifetimes under Type-I, and Type-II censored sampling situations. Moreover, the existence of the unrestricted MLE for a certain stress level is conditional on observing failures at that particular stress level. Under the order restriction, MLE exist even for stress levels without observed failures, provided that these stress levels are internal. We also discuss hypothesis testing problems under order restrictions.      

Optimum Bivariate Step-Stress Accelerated Life Test for Censored Data

A step-stress accelerated life test for two stress variables is developed. The time to failure follows the Weibull distribution, and the test is subject to termination at a predetermined time, leading to censored failure data. An optimum test plan is developed to determine the test interval for each combination of stress levels. The scale parameter of the Weibull distribution for each combination of stress levels is defined as a log linear function of the stress levels. The optimal criterion is defined to minimize the asymptotic variance of the maximum likelihood estimator of the life for a specified reliability     

Thresholding method for dimensionality reduction in recognitionsystems

Often recognition systems must be designed with a relatively small amount of training data. Plug-in test statistics suffer from large estimation errors, often causing the performance to degrade as the measurement vector dimension increases. Choosing a better test statistic or applying a method of dimensionality reduction are two possible solutions to this problem. In this paper, we consider a recognition problem where the data for each population are assumed to have the same parametric distribution but differ in their unknown parameters. The collected vectors of data as well as their components are assumed to be independent. The system is designed to implement a plug-in log-likelihood ratio test with maximum-likelihood (ML) estimates of the unknown parameters instead of the true parameters. Because a small amount of data is available to estimate the parameters, the performance of such a system is strongly degraded relative to the performance with known parameters. To improve the performance of the system we define a thresholding function that, when incorporated into the plug-in log-likelihood ratio, significantly decreases the probability of error for binary and multiple hypothesis testing problems for the exponential class of populations. We analyze the modified test statistic and present the results of Monte Carlo simulation. Special attention is paid to the complex Gaussian model with zero mean and unknown variances     

Testing Exponentiality Based on Kullback-Leibler Information With Progressively Type-II Censored Data

We express the joint entropy of progressively censored order statistics in terms of an incomplete integral of the hazard function, and provide a simple estimate of the joint entropy of progressively Type-II censored data. We then construct a goodness-of-fit test statistic based on Kullback-Leibler information with progressively Type-II censored data. Finally, by using Monte Carlo simulations, the power of the test is estimated, and compared against several alternatives under different progressive censoring schemes     

指数分步场合分组数据下简单步加试验的极大似然估计

对指数分布场合下的寿命分布进行研究,简单步进应力加速寿命试验中,在观察时间间隔不相等的条件下,对所获得的分组数据为寿命数据,给出加速方程中未知参数的极大似然估计（MLE）,从而计算出常应力下产品的平均寿命。     

真空荧光显示屏恒定及步进应力加速寿命试验的研究

为了解决在较短的时间内预测真空荧光显示屏（VFD）寿命的问题,降低寿命预测成本,通过加大灯丝温度进行了恒定和步进应力相组合的加速寿命试验,研究制定了其加速寿命试验的设计方案。应用威布尔分布函数描述其寿命分布,利用最小二乘法完成了试验数据的统计和分析,并开发了寿命预测软件。研究结果表明．试验设计方案是正确可行的,VFD的寿命服从威布尔分布,其加速模型符合阿伦尼斯方程,加速参数的精确计算确保以后在很短的时间内便可估算出VFD在正常应力下的寿命。     

On the asymptotic properties of a nonparametric L/sub 1/-test statistic of homogeneity

We present two simple and explicit procedures for testing homogeneity of two independent multivariate samples of size n. The nonparametric tests are based on the statistic T/sub n/, which is the L/sub 1/ distance between the two empirical distributions restricted to a finite partition. Both tests reject the hypothesis of homogeneity if T/sub n/ becomes large, i.e., if T/sub n/ exceeds a threshold. We first discuss Chernoff-type large deviation properties of T/sub n/. This results in a distribution-free strong consistent test of homogeneity. Then the asymptotic distribution of the test statistic is obtained, leading to an asymptotically /spl alpha/-level test procedure.     

对数正态分布下基于MAM的VFD加速寿命试验研究

为了在较短时间内掌握真空荧光显示器(VFD)的寿命信息,节省寿命试验时间,通过开展两组恒定应力加速寿命试验和一组步进应力加速寿命试验,采用对数正态分布函数描述VFD的寿命分布,在图分析法(MAM)的基础上,基于MATLAB软件强大的计算和可视化绘图功能,绘制了对数正态概率双坐标纸,估计了对数均值和对数标准方差,完成了VFD恒定应力和步进应力加速寿命试验数据的统计和分析.结果表明,VFD寿命服从对数正态分布,其加速模型符合阿伦尼斯方程,基于MATLAB的图分析法来实现寿命数据处理是可行的,精确估算的加速参数使得快速预测VFD寿命成为可能.