Simple Step-stress Test Research Articles

Designing efficient Bayesian sampling plans for exponential distributions based on samples under a (n, t) simple step-stress test

This article studies a method about how to design an efficient Bayesian sampling plan based on samples collected through a (n, t)-sampling scheme with a simple step-stress accelerated life test (ALT). For brevity, such a step-stress accelerated life test is called (n, t)-SSALT. A Bayesian sampling plan (BSP) derived through the (n, t)-SSALT is called BSPA. First, we derive the BSPA with data collected through a (n, t)-SSALT for a general loss function. Given gamma and Jeffreys prior distributions, an explicit expression of the Bayes decision function under a certain loss function is derived. By applying a curtailment procedure to the preceding Bayes decision function, an on-line new Bayes decision function and Bayesian sampling plan, called an efficient Bayesian sampling plan (EBSPA), are constructed. It is shown that the Bayes risk of EBSPA is less than or equal to that of BSPA. This indicates that the EBSPA is more efficient than the BSPA. Comparisons among some BSPAs and the proposed EBSPA are given. The numerical results indicate that, in terms of Bayes risks, the EBSPA significantly outperforms the BSPA.

Designing efficient Bayesian sampling plans based on the simple step-stress test under random stress-change time for censored data

ABSTRACT This paper studies the problem about how to design an efficient Bayesian sampling plan based on a simple step-stress test with a random stress-change time for censored data. For brevity, the step-stress test with random stress-change time is briefly called SSRSCT. A Bayesian sampling plan (BSP) through the SSRSCT is called BSPA. First, we derive the BSPA with data based on Type-II censoring for a general loss function. Given gamma and noninformative uniform prior distributions, an explicit expression of the Bayes decision function under a certain loss function is derived. By applying a curtailment procedure to the preceding Bayes decision function, a new Bayesian sampling plan, called an efficient Bayesian sampling plan (EBSPA), is constructed. It is shown that the Bayes risk of EBSPA is less than or equal to that of BSPA. This indicates that EBSPA is more efficient than the BSPA. Comparisons among some BSPAs and the proposed EBSPA are given. Monte Carlo simulations are carried out, and results indicate that the EBSPA outperforms those BSPAs. The EBSPA is recommended.

K‐step stage life testing

The model of stage life testing proposed in Laumen and Cramer (2019b) is extended from two to k stages. It is illustrated that this model can be seen as an extension of progressive censoring with fixed censoring times as well as of simple step stress testing. Extending the first model, the new approach allows to incorporate information from progressively censored units subject to additional life testing. On the other hand, simple step stress modeling is generalized in the sense that only parts of the units under test are put on another stress level whereas the others remain on the initial level. The model is further studied for exponential lifetimes assuming a cumulative exposure model. We obtain the exact (conditional) distribution of the maximum likelihood estimators which is applied to construct (exact) confidence intervals for the distribution parameters. Finally, we investigate the situation of Weibull distributed lifetimes on the initial stage. The study is supplemented by illustrative simulations.

A Note on the Prediction of Censored Exponential Lifetimes in a Simple Step-stress Model with Type-II Censoring

In this article, we consider the problem of predicting survival times of units from the exponential distribution which are censored under a simple step-stress testing experiment. In this article, Type-II censoring are considered for the form of censoring. Two kinds of predictors—the maximum likelihood predictors (MLP) and the conditional median predictors (CMPs)—are derived. Some numerical examples are presented to illustrate the prediction methods developed here. Using simulation studies, prediction intervals are generated for these examples. We then compare the MLP and the CMP with respect to mean squared prediction error (MSPE) and the prediction interval. Finally, we used a real data to apply the prediction methods developed in the article.

Joint modeling of linear degradation and failure time data with masked causes of failure under simple step-stress test

In this paper, we propose a method to model the relationship between degradation and failure time for a simple step-stress test where the underlying degradation path is linear and different causes of failure are possible. It is assumed that the intensity function depends only on the degradation value. No assumptions are made about the distribution of the failure times. A simple step-stress test is used to induce failure experimentally and a tampered failure rate model is proposed to describe the effect of the changing stress on the intensities. We assume that some of the products that fail during the test have a cause of failure that is only known to belong to a certain subset of all possible failures. This case is known as masking. In the presence of masking, the maximum likelihood estimates of the model parameters are obtained through the expectation–maximization algorithm by treating the causes of failure as missing values. The effect of incomplete information on the estimation of parameters is studied through a Monte-Carlo simulation. Finally, a real-world example is analysed to illustrate the application of the proposed methods.

Statistical inference for masked interval data with Weibull distribution under simple step-stress test and tampered failure rate model

In this article, we consider the planning of simple step-stress life test in the presence of competing risks with the possibility of masked causes of failure and interval censoring. It is assumed t...

Prediction of censored exponential lifetimes in a simple step-stress model under progressive Type II censoring

In this article, we consider the problem of predicting survival times of units from the exponential distribution which are censored under a simple step-stress testing experiment. Progressive Type-II censoring are considered for the form of censoring. Two kinds of predictors—the maximum likelihood predictors (MLP) and the conditional median predictors (CMP)—are derived. Some numerical examples are presented to illustrate the prediction methods developed here. Using simulation studies, prediction intervals are generated for these examples. We then compare the MLP and the CMP with respect to mean squared prediction error and the prediction interval.

Simple step-stress model for an extension of the exponential distribution with type-I censoring

Purpose– The purpose of this paper is to design a simple step-stress model under type-I censoring when the failure time has an extension of the exponential distribution.Design/methodology/approach– The scale parameter of the distribution is assumed to be a log-linear function of the stress and a cumulative exposure model is hold. The maximum likelihood estimates of the parameters, as well as the corresponding Fisher information matrix are derived. Two real examples are given to show the application of an extension of the exponential distribution in reliability studies and a numerical example is presented to illustrate the method discussed here.Findings– A simple step-stress test under cumulative exposure model and type-I censoring for an extension of the exponential distribution is presented.Originality/value– The work is original.

Minimal repair under a step-stress test

In the one- and multi-sample cases, in the context of life-testing reliability experiments, we introduce minimal repair processes under a simple step-stress test, based on exponential distributions and an associated cumulative exposure model, and then develop likelihood inference for such a model.

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.

Step-stress life-testing with random stress-change times for exponential 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 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 simple step-stress accelerated life tests with censoring

The authors present the optimum simple time-step and failure-step stress accelerated life tests for the case where a prespecified censoring time is involved. An exponential life distribution with a mean that is a log-linear function of stress, and a cumulative exposure model are assumed. The authors obtain the optimum test plans to minimize the asymptotic variance of the maximum-likelihood estimator of the mean life at a design stress. Nomographs for the optimum time-step stress test are also given. Simple step stress test plans with censoring were studied. The optimum failure-step stress test plans are obtained in a closed form, whereas, for the time-step stress test, nomographs are given for finding the optimum plans on the basis of parameters. The parameters must be approximated from experience, similar data, or a preliminary test. For some selected values of the parameters, the effect of incorrect preestimates was studied in terms of the percentage of variance increase. The effect was found to be small. >

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.