Analysis of a simple step-stress life test with a random stress-change time

This paper studies a variation of a simple step-stress life testing in which the stress change time is random, and the test is subject to type II censoring. We assume that only two order statistics from the test are observed. The first observed order statistic is the stress change time from a low level stress to a high level stress during the testing, and the second observed order statistic is the final failure time when the test is censored. We first present the joint probability distribution of the two order statistics observed from the simple step-stress accelerated life test. Maximum likelihood estimates, and the method of moment estimates for model parameters based on the joint distribution are considered. We also present the exact confidence interval estimates for the model parameters based on various pivotal quantities, and demonstrate the estimation procedure by a simulated example.     

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     

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     

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.     

Commentary: inference in simple step-stress models

This note considers simple step-stress models with type-II censored exponential data. For two stress-levels and a log-linear link between stress and mean life, there is a reparameterization of the parameters of the negative exponential distributions, so that one can equally well work with these parameters or those in the log-linear link. From a statistical perspective, it is preferable to work with the parameters of the negative exponential distributions, since the "lack-of-memory" property of this distribution allows the analysis to be sub-divided, with ensuing simplification     

Estimation of Reliability from Multiple Independent Grouped Censored Samples with Failure Times Known

Failure times of one type aircraft-engine component were recorded. In addition, life times are periodically recorded for unfailed engine components. The data are considered as multiple s-independent grouped censored samples with failure times known. The assumed failure model is the 2-parameter Weibull distribution. Maximum likelihood estimates are derived. The exponential model is used for comparison. Monte Carlo simulation is used to derive s-bias and mean square error of the estimates. The asymptotic covariance matrix was computed for the sampling conditions studied. The maximum likelihood estimates of the reliability were obtained as a function of component operating time since overhaul.     

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。     

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.     

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 simple step-stress accelerated life-tests with competingcauses of failure

Optimum simple step-stress accelerated life tests (ALTs) for products with competing causes of failure are presented. The life distribution of each failure cause, which is independent of the others, is assumed to be exponential with a mean that is a log-linear function of the stress, and a cumulative exposure model is assumed. Optimum plans for time-step and failure-step ALTs are obtained which minimize the sum over all failure causes of asymptotic variances of the maximum likelihood estimators of the log mean lives at design stress. The competing causes of failure affect the optimum test plan only through the product of two ratios-the ratio of the sums of the mean lives and the ratio of the sums of the failure rates over all failure causes at low and high stress levels. The effect of this product (of two ratios) is studied     

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.     

Test exponentiality against a monotone hazard function alternative based on TTT transformation

This note introduces a new method to test exponentiality against the monotone hazard function alternative based on the total time on test. The proposed statistics are similar to L-statistics, so the asymptotic normality can be established by using the asymptotic normality of L-statistics. As a remark, the modified version of the test is proposed to accommodate the type II censored data. In the small sample case the quantile points have been obtained by using the Monte Carlo method. We also examine the power test against the single parameter Weibull model. As a remark we make a comparison with the well known and efficient Gail and Gastwirth test.     

A new model for step-stress testing

The mathematical intractability of the Weibull cumulative exposure model (CE-M) has impeded the development of statistical procedures for step-stress accelerated life tests. Our new model (KH-M) is based on a time transformation of the exponential CE-M. The time-transformation enables the reliability engineer to use known results for multiple-step, multiple-stress models that have been developed for the exponential step-stress model. KH-M has a realistically appealing proportional-hazard property. It is as flexible as the Weibull CE-M for fitting data, but its mathematical form makes it easier to obtain parameter estimates and standard deviations. Maximum likelihood estimates are given for test plans with unknown shape parameter. The mathematical similarity to the constant-stress Weibull model is shown. Chi-square goodness of fit tests are performed on simulated data to compare the fit of the models     

Estimation of Periodically Changing Failure Rate

This paper considers two life testing procedures (progressively censored samples and Bartholomew's experiment) under the assumption that the life of an item follows the exponential distribution. The failure rates are different under n different conditions of usage of the item at regular intervals of time. The maximum likelihood estimates of the n failure rates have been derived along with their asymptotic variances for both types of data (when failure times are recorded and when only the number of items failing in each interval are recorded). A numerical example illustrates the type of data and relevant calculations for the experiment involving progressively censored samples.     

Estimation of threshold stress in accelerated life-testing

The author presents a method that uses accelerated life-test data to estimate the mean life at the service stress and the threshold stress below which a failure is unlikely to occur. The relation between stress and mean-life at that stress is assumed to follow an inverse power law that includes a threshold stress. The failure times at a given stress are assumed to follow a Weibull distribution in which the shape parameter varies with the stress. This model extends the well-known Weibull inverse power law model. If only the mean life but not a specific percentile point at a service stress is sought, the maximum likelihood method is useful for parameter estimation. This is a tradeoff in the parametric approach. For adoption of an appropriate probability model, the likelihood ratio test as well as the Akaike Information Criterion are used. Type I right censored data are considered. Extensions of the method are discussed     

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.     

On Testing Monotonicity of Mean Residual Life from Randomly Censored Data

This paper proposes a new nonparametric test for testing the null hypothesis that the MRL is constant against the alternative hypothesis that the MRL is decreasing (increasing) for ramdomly censored data. The proposed test statistic is a L-statistic, and we use L-statistic theory to establish its asymptotic normality of the test statistic. We discuss the efficiency loss due to censoring and also calculate the asymptotic relative efficiencies of our test statistic with respect to the Chen, Hollander and Langberg's test for several alternatives.     

Accelerated life tests analyzed by a piecewise exponentialdistribution via generalized linear models

Efficient industrial experiments for the reliability analysis of manufactured products consist of subjecting the units to accelerated life tests where, for each pre-fixed stress level, the experiment ends after the failure of a certain pre-fixed proportion of units or a certain test time is reached. This paper estimates the mean life of the units under usual working conditions, based on censored data obtained from units under stress conditions. This problem is approached through a generalized linear model and related inferential techniques, considering the very flexible class of failure distributions, piecewise exponential model and a log-linear stress-response relationship. The general framework has as particular cases, among others, the power law model, the Arrhenius model and the generalized Eyring model. A numerical example illustrates the methodology     

Confidence interval for the mean of the exponential distribution,based on grouped data

When the available data from an exponential distribution are grouped, the maximum likelihood estimator (MLE) for the mean and several modified MLE have been discussed in literature. However, little work has been done on interval estimators based on such grouped data. This paper derives the asymptotic property of a statistic which is used to construct an approximate confidence interval for the mean. The width of this approximate confidence interval, based on grouped data, is compared with those based on complete samples, and samples with type-I and type-II censoring. The limits of the ratios of these widths are derived when the sample size approaches infinity. The approximate confidence interval from grouped data is wider than those from complete and censored samples. However, Monte Carlo simulation indicates that the proposed method based on grouped data is adequate, considering the restricted information in this case     

Optimum simple ramp-tests for the Weibull distribution and type-Icensoring

An optimum simple ramp test-accelerated life test with two different linearly increasing stresses-is presented for the Weibull distribution under type I censoring. It is assumed that the inverse power law holds between the Weibull scale parameter and the constant stress and that the cumulative exposure model for the effect of changing stress applies. The optimum plan-low stress rate and proportion of test units allocated to low stress mode-is found. It minimizes the asymptotic variance of the maximum likelihood estimator of a stated quantile at design stress. For selected values of the design parameters, these optimum plans are tabulated, and the effect of the preestimates of these parameters are studied