Optimal step-stress accelerated degradation test plans for inverse Gaussian process based on proportional degradation rate model

The purpose of this paper is to address the optimal design of the step-stress accelerated degradation test (SSADT) issue when the degradation process of a product follows the inverse Gaussian (IG) process. For this design problem, an important task is to construct a link model to connect the degradation magnitudes at different stress levels. In this paper, a proportional degradation rate model is proposed to link the degradation paths of the SSADT with stress levels, in which the average degradation rate is proportional to an exponential function of the stress level. Two optimization problems about the asymptotic variances of the lifetime characteristics' estimators are investigated. The optimal settings including sample size, measurement frequency and the number of measurements for each stress level are determined by minimizing the two objective functions within a given budget constraint. As an example, the sliding metal wear data are used to illustrate the proposed model.     

A meta-analysis approach for step-stress experiments

The step-stress model is a special case of accelerated life testing that allows for testing of units under different levels of stress with changes occurring at various intermediate stages of the experiment. Interest then lies on inference for the mean lifetime at each stress level. All step-stress models discussed so far in the literature are based on a single experiment. For the situation when data have been collected from different experiments wherein all the test units had been exposed to the same levels of stress but with possibly different points of change of stress, we introduce a model that combines the different experiments and facilitates a meta-analysis for the estimation of the mean lifetimes. We then discuss in detail the likelihood inference for the case of simple step-stress experiments under exponentially distributed lifetimes with Type-II censoring.     

Optimal Design for Step-Stress Accelerated Degradation Test with Multiple Performance Characteristics Based on Gamma Processes

Step-stress accelerated degradation test (SSADT) plays an important role in assessing the lifetime distribution of highly reliable products under normal operating conditions when there are not enough test units available for testing purposes. Recently, the optimal SSADT plans are presented based on an underlying assumption that there is only one performance characteristic. However, many highly reliable products usually have complex structure, with their reliability being evaluated by two or more performance characteristics. At the same time, the degradation of these performance characteristics would be always positive and strictly increasing. In such a case, the gamma process is usually considered as a degradation process due to its independent and nonnegative increments properties. Therefore, it is of great interest to design an efficient SSADT plan for the products with multiple performance characteristics based on gamma processes. In this work, we first introduce reliability model of the degradation products with two performance characteristics based on gamma processes, and then present the corresponding SSADT model. Next, under the constraint of total experimental cost, the optimal settings such as sample size, measurement times, and measurement frequency are obtained by minimizing the asymptotic variance of the estimated 100 qth percentile of the product’s lifetime distribution. Finally, a numerical example is given to illustrate the proposed procedure.     

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.     

Asymptotic Comparison Between Constant-stress Testing and Step-stress Testing for Type-I Censored Data from Exponential Distribution

By running the life tests at higher stress levels than normal operating conditions, accelerated life testing 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 the test, while step-stress testing allows the experimenter to gradually increase the stress levels at some pre-fixed time points during the test. In this article, the optimal k-level constant-stress and step-stress accelerated life tests are compared for the exponential failure data under Type-I censoring. The objective is to quantify the advantage of using the step-stress testing relative to the constant-stress one. A log-linear relationship between the mean lifetime parameter and stress level is assumed and the cumulative exposure model holds for the effect of changing stress in step-stress testing. The optimal design point is then determined under C-optimality, D-optimality, and A-optimality criteria. The efficiency of step-stress testing compared to constant-stress testing is discussed in terms of the ratio of optimal objective functions based on the information matrix.     

Exact inference for a simple step-stress model with competing risks for failure from exponential distribution under Type-II censoring

In reliability analysis, accelerated life-testing allows for gradual increment of stress levels on test units during an experiment. In a special class of accelerated life tests known as step-stress tests, the stress levels increase discretely at pre-fixed time points, and this allows the experimenter to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. Moreover, when a test unit fails, there are often more than one fatal cause for the failure, such as mechanical or electrical. In this article, we consider the simple step-stress model under Type-II censoring when the lifetime distributions of the different risk factors are independently exponentially distributed. Under this setup, we derive the maximum likelihood estimators (MLEs) of the unknown mean parameters of the different causes under the assumption of a cumulative exposure model. The exact distributions of the MLEs of the parameters are then derived through the use of conditional moment generating functions. Using these exact distributions as well as the asymptotic distributions and the parametric bootstrap method, we discuss the construction of confidence intervals for the parameters and assess their performance through Monte Carlo simulations. Finally, we illustrate the methods of inference discussed here with an example.     

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.     

Bivariate Constant-Stress Accelerated Degradation Model and Inference

To assess the reliability of highly reliable products that have two or more performance characteristics (PCs) in an accurate manner, relations between the PCs should be taken duly into account. If they are not independent, it would then become important to describe the dependence of the PCs. For many products, the constant-stress degradation test cannot provide sufficient data for reliability evaluation and for this reason, accelerated degradation test is usually performed. In this article, we assume that a product has two PCs and that the PCs are governed by a Wiener process with a time scale transformation, and the relationship between the PCs is described by the Frank copula function. The copula parameter is dependent on stress and assumed to be a function of stress level that can be described by a logistic function. Based on these assumptions, a bivariate constant-stress accelerated degradation model is proposed here. The direct likelihood estimation of parameters of such a model becomes analytically intractable, and so the Bayesian Markov chain Monte Carlo (MCMC) method is developed here for this model for obtaining the maximum likelihood estimates (MLEs) efficiently. For an illustration of the proposed model and the method of inference, a simulated example is presented along with the associated computational results.     

Order restricted Bayesian inference for exponential simple step-stress model

A step-stress model has received a considerable amount of attention in recent years. In the usual step-stress experiment, a stress level is allowed to increase at each step to get rapid failure of the experimental units. The expected lifetime of the experimental unit is shortened as the stress level increases. Although extensive amount of work has been done on step-stress models, not enough attention has been paid to analyze step-stress models incorporating this information. We consider a simple step-stress model and provide Bayesian inference of the unknown parameters under cumulative exposure model assumption. It is assumed that the lifetime of the experimental units are exponentially distributed with different scale parameters at different stress levels. It is further assumed that the stress level increases at each step, hence the expected lifetime decreases. We try to incorporate this restriction using the prior assumptions. It is observed that different censoring schemes can be incorporated very easily under a general setup. Monte Carlo simulations have been performed to see the effectiveness of the proposed method, and two datasets have been analyzed for illustrative purposes.     

Optimal design of accelerated degradation tests based on Wiener process models

Optimal accelerated degradation test (ADT) plans are developed assuming that the constant-stress loading method is employed and the degradation characteristic follows a Wiener process. Unlike the previous works on planning ADTs based on stochastic process models, this article determines the test stress levels and the proportion of test units allocated to each stress level such that the asymptotic variance of the maximum-likelihood estimator of the qth quantile of the lifetime distribution at the use condition is minimized. In addition, compromise plans are also developed for checking the validity of the relationship between the model parameters and the stress variable. Finally, using an example, sensitivity analysis procedures are presented for evaluating the robustness of optimal and compromise plans against the uncertainty in the pre-estimated parameter value, and the importance of optimally determining test stress levels and the proportion of units allocated to each stress level are illustrated.     

Failure Inference and Optimization for Step Stress Model Based on Bivariate Wiener Model

In this article, we consider the situation under a life test, in which the failure time of the test units are not related deterministically to an observable stochastic time varying covariate. In such a case, the joint distribution of failure time and a marker value would be useful for modeling the step stress life test. The problem of accelerating such an experiment is considered as the main aim of this article. We present a step stress accelerated model based on a bivariate Wiener process with one component as the latent (unobservable) degradation process, which determines the failure times and the other as a marker process, the degradation values of which are recorded at times of failure. Parametric inference based on the proposed model is discussed and the optimization procedure for obtaining the optimal time for changing the stress level is presented. The optimization criterion is to minimize the approximate variance of the maximum likelihood estimator of a percentile of the products’ lifetime distribution.     

Optimum M-step,step-stress design with K stress variables

Most of the available literature on accelerated life testing deals with tests that use only one accelerating variable and no other explanatory variables. Frequently, however, there is a need to use more than one accelerating or other experimental variables. Examples include a test of capacitors at higher than usual levels of temperature and voltage, and a test of circuit boards at higher than usual levels of temperature, humidity, and voltage. M-step, step-stress models are extended to include k stress variables. Optimum M-step, step-stress designs with k stress variables are found. The polynomial model is considered as a special case, and a lack of fit test is discussed. Also a goodness-of-fit test is proposed and the appropriateness of using its asymptotic chi-square distribution for small samples is shown.     

Analysis of simple step-stress model in presence of competing risks

ABSTRACT

In this article, we consider a simple step-stress life test in the presence of exponentially distributed competing risks. It is assumed that the stress is changed when a pre-specified number of failures takes place. The data is assumed to be Type-II censored. We obtain the maximum likelihood estimators of the model parameters and the exact conditional distributions of the maximum likelihood estimators. Based on the conditional distribution, approximate confidence intervals (CIs) of unknown parameters have been constructed. Percentile bootstrap CIs of model parameters are also provided. Optimal test plan is addressed. We perform an extensive simulation study to observe the behaviour of the proposed method. The performances are quite satisfactory. Finally we analyse two data sets for illustrative purposes.     

Partially Accelerated Life Tests for the Weibull Distribution Under Multiply Censored Data

This article aims to estimate the parameters of the Weibull distribution in step-stress partially accelerated life tests under multiply censored data. The step partially acceleration life test is that all test units are first run simultaneously under normal conditions for a pre-specified time, and the surviving units are then run under accelerated conditions until a predetermined censoring time. The maximum likelihood estimates are used to obtaining the parameters of the Weibull distribution and the acceleration factor under multiply censored data. Additionally, the confidence intervals for the estimators are obtained. Simulation results show that the maximum likelihood estimates perform well in most cases in terms of the mean bias, errors in the root mean square and the coverage rate. An example is used to illustrate the performance of the proposed approach.     

Accelerated Degradation Models for Failure Based on Geometric Brownian Motion and Gamma Processes

Based on a generalized cumulative damage approach with a stochastic process describing degradation, new accelerated life test models are presented in which both observed failures and degradation measures can be considered for parametric inference of system lifetime. Incorporating an accelerated test variable, we provide several new accelerated degradation models for failure based on the geometric Brownian motion or gamma process. It is shown that in most cases, our models for failure can be approximated closely by accelerated test versions of Birnbaum–Saunders and inverse Gaussian distributions. Estimation of model parameters and a model selection procedure are discussed, and two illustrative examples using real data for carbon-film resistors and fatigue crack size are presented.     

Weibull lagged effect step-stress model with competing risks

In many survival analysis studies, failure can come from one of several competing risks. Additionally, where survival times are lengthy, researchers can increase stress levels to cause units to fail faster. One type of accelerated testing is a step-stress test where the increase is presented in quantum jumps at predetermined time points. If the impact of the increase is not immediately attained, an interim lag period is modeled. In this article, we propose a two-competing risk step-stress model with a lag period where each independent risk follows a Weibull lifetime distribution, the interim lag period is linear, and the attainment point is assumed known. We obtain the maximum likelihood estimators and the observed information matrix; we construct confidence intervals and provide estimates of coverage probabilities using large sample theory, percentile bootstrap, and bias-corrected accelerated (BCa) bootstrap methods.     

Planning step-stress test under Type-I censoring for the exponential case

We consider in this work a k-level step-stress accelerated life-test (ALT) experiment with unequal duration steps τ=(τ₁, …, τ_k). Censoring is allowed only at the change-stress point in the final stage. An exponential failure time distribution with mean life that is a log-linear function of stress, along with a cumulative exposure model, is considered as the working model. The problem of choosing the optimal τ is addressed using the variance-optimality criterion. Under this setting, we then show that the optimal k-level step-stress ALT model with unequal duration steps reduces just to a 2-level step-stress ALT model.     

Exact inference for a simple step-stress model with Type-I hybrid censored data from the exponential distribution

In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress factors such as temperature, voltage and load on the lifetimes of experimental units. Step-stress test, which is a special class of accelerated life-tests, allows the experimenter to increase the stress levels at fixed times during the experiment in order to obtain information on the parameters of the life distributions more quickly than under normal operating conditions. In this paper, we consider the simple step-stress model under the exponential distribution when the available data are Type-I hybrid censored. We derive the maximum likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model with lifetimes being exponentially distributed. The exact distributions of the MLEs of parameters are obtained through the use of conditional moment generating functions. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs and the parametric bootstrap methods, and assess their performance through a Monte Carlo simulation study. Finally, we present two examples to illustrate all the methods of inference discussed here.     

A tampered Brownian motion process model for partial step-stress accelerated life testing

In partial step-stress accelerated life testing, models extrapolating data obtained under more severe conditions to infer the lifetime distribution under normal use conditions are needed. Bhattacharyya (Invited paper for 46th session of the ISI, 1987) proposed a tampered Brownian motion process model and later derived the probability distribution from a decay process perspective without linear assumption. In this paper, the model is described and the features of the failure time distribution are discussed. The maximum likelihood estimates of the parameters in the model and their asymptotic properties are presented. An application of models for step-stress accelerated life test to fields other than engineering is described and illustrated by applying the tampered Brownian motion process model to data taken from a clinical trial.     

Bayesian Analysis of Masked Data in Step-stress Accelerated Life Testing

This article considers a k level step-stress accelerated life testing (ALT) on series system products, where independent Weibull-distributed lifetimes are assumed for the components. Due to cost considerations or environmental restrictions, causes of system failures are masked and type-I censored observations might occur in the collected data. Bayesian approach combined with auxiliary variables is developed for estimating the parameters of the model. Further, the reliability and hazard rate functions of the system and components are estimated at a specified time at use stress level. The proposed method is illustrated through a numerical example based on two priors and various masking probabilities.