Step-stress Accelerated Life Testing Research Articles

Planning step-stress life-test with a target acceleration-factor

A sequential approach is presented to plan a multiple-steps step-stress accelerated life test (SSALT) with type-I censoring so as to achieve a pre-specified acceleration factor. An initial optimal plan for a simple SSALT, where one not only has to determine the optimum hold-time under low stress but also the optimum low stress level, is obtained by solving a constrained nonlinear program. A backward recursion scheme generates the subsequent optimal low-stress levels and hold-times for multiple-step SSALT. An illustration using a 3-step SSALT is presented. The relative efficiency of the 2-step and 3-step SSALT is investigated. A numerical example illustrates the method under 2-step and 3-step SSALT.

Optimal Designs of Simple Step-Stress Accelerated Life Tests for Lognormal Lifetime Distributions

This paper considers optimal designs of step-stress accelerated life tests in which each lognormally-distributed test item is first run at low stress, and if it does not fail for a specified time, then it is run at high stress until a predetermined censoring time. It is assumed that a log-linear relation exists between the lognormal location parameter and stress, and that a cumulative exposure model for the effect of changing stress holds. The optimum stress change point minimizes the asymptotic variance of maximum likelihood estimator of a specified percentile at design stress. For selected values of the design parameters, the optimum plans are tabulated. Designs of high-to-low step-stress accelerated life tests (ALTs) in which each item is first run at high stress and then at low stress, and the optimality criterion of minimizing the generalized asymptotic variance of maximum likelihood estimators of model parameters, are also considered. The effects of the incorrect pre-estimates of the design parameters are investigated.

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.

Optimal design of accelerated life tests under modified stress loading methods

Summary Most of the previous work on optimal design of accelerated life test (ALT) plans has assumed instantaneous changes in stress levels, which may not be possible or desirable in practice, because of the limited capability of test equipment, possible stress shocks or the presence of undesirable failure modes. We consider the case in which stress levels are changed at a finite rate, and develop two types of ALT plan under the assumptions of exponential lifetimes of test units and type I censoring. One type of plan is the modified step-stress ALT plan, and the other type is the modified constant-stress ALT plan. These two plans are compared in terms of the asymptotic variance of the maximum likelihood estimator of the log mean lifetime for the use condition (i.e. avar\\[ln (0)]). Computational results indicate that, for both types of plan, avar\\[ln (0)] is not sensitive to the stress-increasing rate R, if R is greater than or equal to 10, say, in the standardized scale. This implies that the proposed stress loading method can be used effectively with little loss in statistical efficiency. In terms of avar\\[ln (0)], the modified step-stress ALT generally performs better than the modified constant-stress ALT, unless R or the probability of failure until the censoring time under a certain stress-increasing rate is small. We also compare the progressive-stress ALT plan with the above two modified ALT plans in terms of avar\\[ln (0)], using the optimal stress-increasing rate R* determined for the progressivestress ALT plan. We find that the proposed ALTs perform better than the progressivestress ALT for the parameter values considered.

A new way to estimate the parameters in the progressive stress accelerated life testing

Among the three types of accelerated life testing, that is, the constant stress accelerated life testing, the step stress accelerated life testing, and the progressive stress accelerated life testing, the last one is the most effective and economical way to get the failure times of products in a short period of time. In this article a deficiency in the classical approach is pointed out and a new method for the statistical analysis of life data, especially for electronic products, under progressive stress (voltage)V(t) =Kt is given. Using the Bayesian method, for certain choices of the prior distribution, several forms of estimators of the parameters in the normal stress Weibull distribution and the inverse power law model are derived. In particular, when the new shape parameter of the distribution under progressive stress is given (or estimated in advance), the closed forms of these estimators are relatively simple and are numerically accessible on computers. The formulations are given for the general case and are illustrated for a special simple case. Comparison with the classical approach using pseudorandom data as well as real data on solid tantatum electrolytic capacitors shows that the proposed method is better and more effective.

Nonparametric model for step-stress accelerated life testing

A nonparametric model of the interval estimation is proposed. The estimation is based on results of step-stress accelerated life testing. The approach includes: application of a linear model of damage accumulation, which is proved to be equivalent to the assumptions accepted in numerous papers; estimation of the time transformation function, using the test results for all the load levels; and application of a nonparametric approach for interval estimation of reliability measures, based on the assumption that life-time distribution under the constant load has increasing failure rate. A numerical example uses initial data obtained on the basis of statistical simulation thus verifying the proposed approach.

Optimum 3-step step-stress tests

Optimum simple step-stress accelerated life testing plans have practical limitations. They highly depend upon the assumption of a linear relationship between stress and time-to-failure and use only two extreme stresses that can cause irrelevant failure modes. Thus, 3-step plans are preferable. In this paper, the authors derive the optimum quadratic plan and propose and evaluate a 3-step test plan, the compound linear, as an alternative to the optimum simple step-stress plan and optimum quadratic plan. The compound linear plan is better than two compromise plans proposed for constant stress testing and nearly as good as a proposed optimum compromise plan, the advantage being its simplicity. A small sample-size study shows that the efficiency of the test plan is not appreciably affected by the sample size, so that the asymptotic results are a good indication of the performance in sample sizes encountered in practice.

A Bayes approach to step-stress accelerated life testing

This paper develops a Bayes model for step-stress accelerated life testing. The failure times at each stress level are exponentially distributed, but strict adherence to a time-transformation function is not required. Rather, prior information is used to define indirectly a multivariate prior distribution for the failure rates at the various stress levels. Our prior distribution preserves the natural ordering of the failure rates in both the prior and posterior estimates. Methods are developed for Bayes point estimates as well as for making probability statements for use-stress life parameters. The approach is illustrated with an example.

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.

Multiple step-stress accelerated life test: the tampered failure rate model

Battacharyya and Soejoeti (1989) proposed the tampered failure rate model for step-stress accelerated life testing. In this note, their model is generalized from the simple (2-step) step-stress setting to the multiple (k-step, k > 2) setting. For the parametric setting where the life distribution under constant stress is Weibull, maximum likelihood estimation is investigated and the situation where the different stress levels are equispaced is looked at.

Equivalence of the tampered random variable and the tampered failure rate models in accelerated lifetesting for a class of life distributions having the ‘setting the clock back to zero property'

It is well-known that Nelson's (1980) Cumulative Exposure (CE) model in a step-stress accelerated life test is equivalent to the Tampered Random Variable (TRV) model of DeGroot and Goel (1979) if the life distributions under the two stresses belong to the same scale-parameter family. In the present paper, it is proved that if a step-stress accelerated life test is performed on a population of units whose life distribution belongs to a class of life distributions having the ‘setting the clock back to zero’ property, (see B.R. Rao and Talwaker 1989, 1990 and B.R. Rao 1990a), then the TRV is equivalent to the Tampered Failure Rate (TFR) model of Bhattacharyya and Soejoeti (1989). The proof of the necessity is simple. Simple examples of the family of life distributions with this property include the exponential, the linear hazard exponential and the Gompertz distributions. The Weibull distribution does not have this property and it is demonstrated that the TRV and the TFR models lead to different surival dist...

Optimum simple step-stress accelerated life-tests with competing causes 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.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optimum simple step-stress accelerated life tests with censoring : D. S. Bai, M. S. Kim and S. H. Lee. IEEE Trans. Reliab.38(5), 528 (1989)

Optimum simple step-stress accelerated life tests with censoring : D. S. Bai, M. S. Kim and S. H. Lee. IEEE Trans. Reliab.38(5), 528 (1989)

A tampered failure rate model for step-stress accelerated life test

A statistical model for step-stress accelerated life test is motivated from the point of view that a change of the stress has a multiplicative effect on the failure rate function over the remaining life.Properties of the proposed model, including an interpretation in terms of the conditional reliability, and relationships with the existing models are discussed. For the parametric setting of a Weibull family representing the life distribution under a constant stress, maximum likelihood estimation of the parameters is investigated and the Fisher information matrix is derived. The proposed model is found to have certain analytical advantages over the cumulative exposure model that is commonly used in step-stress analysis. An extension of the model to include a regression structure and inferences for life under the use condition stress are briefly discussed.

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. >

Inference for step-stress accelerated life tests under arbitrary right-censorship

A step-stress test is an accelerated life test in which the stress on an unfailed item is allowed to increase at a preassigned time. Step-stress testing is considered here from a non-parametric point of view when the observations have been arbitrarily right-censored. Thus, no assumptions are made about the particular form of the underlying distribution of life lengths. An estimator of the underlying life distribution under the use conditions stress is obtained. The estimator is shown to be strongly consistent.

Inference for step-stress accelerated life tests

In this paper we consider the more realistic aspect of accelerated life testing wherein the stress on an unfailed item is allowed to increase at a preassigned test time. Such tests are known as step-stress tests. Our approach is nonparametric in that we do not make any assumptions about the underlying distribution of life lengths. We introduce a model for step-stress testing which is based on the ideas of shock models and of wear processes. This model unifies and generalizes two previously proposed models for step-stress testing. We propose an estimator for the life distribution under use conditions stress and show that this estimator is strongly consistent.

Step-stress accelerated life testing of diodes

Step-stress accelerated life testing of diodes

Accelerated life testing and over-stress testing of transistors

The advantages and disadvantages of accelerated life tests compared with conventional life tests are discussed and the methods by which accelerated life tests are used are examined. The physical basis of accelerated life tests in reaction kinetics is described and the methods by which the Arrhenius equation and the Eyring equation can be used to relate time to failure and temperature are discussed. On the basis of the Arrhenius equation and using the log normal failure distribution the results of temperature storage accelerated life tests on germanium alloy transistors are used to make predictions about life behaviour at lower temperatures. These predictions are compared with the results of long-term life tests on several hundred transistors extending to 20,000 hr. The results are then used to compare constant stress and step stress accelerated life tests. Electrical over-stress tests are described in which germanium alloy transistors are subjected to very high power pulses of short duration and silicon planar transistors have high currents passed through their emitter junctions in the reverse direction. Mechanical over-stress tests are exemplified by centrifuge tests on germanium alloy transistors. It is shown for one transistor type that the failure distribution, although in all cases normal, is markedly dependent upon the direction of stress.