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.     

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.     

Exact inference for a simple step-stress model with Type-II 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 a new step-stress model in which the life-testing experiment gets terminated either at a pre-fixed time (say, _Tm+1$T_{m+1}$) or at a random time ensuring at least a specified number of failures (say, r out of n). Under this model in which the data obtained are Type-II hybrid censored, we consider the case of exponential distribution for the underlying lifetimes. We then 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.     

Exponential progressive step-stress life-testing with link function based on Box–Cox transformation

In order to quickly extract information on the life of a product, accelerated life-tests are usually employed. In this article, we discuss a k-stage step-stress accelerated life-test with M-stress variables when the underlying data are progressively Type-I group censored. The life-testing model assumed is an exponential distribution with a link function that relates the failure rate and the stress variables in a linear way under the Box–Cox transformation, and a cumulative exposure model for modelling the effect of stress changes. The classical maximum likelihood method as well as a fully Bayesian method based on the Markov chain Monte Carlo (MCMC) technique is developed for inference on all the parameters of this model. Numerical examples are presented to illustrate all the methods of inference developed here, and a comparison of the ML and Bayesian methods is also carried out.     

Inference for a Simple Step-Stress Model with Type-I Censoring and Lognormally Distributed Lifetimes

Accelerated life-testing (ALT) is a very useful technique for examining the reliability of highly reliable products. It allows the experimenter to obtain failure data more quickly at increased stress levels than under normal operating conditions. A step-stress model is one special class of ALT, and in this article we consider a simple step-stress model under the cumulative exposure model with lognormally distributed lifetimes in the presence of Type-I censoring. We then discuss inferential methods for the unknown parameters of the model by the maximum likelihood estimation method. Some numerical methods, such as the Newton–Raphson and quasi-Newton methods, are discussed for solving the corresponding non-linear likelihood equations. Next, we discuss the construction of confidence intervals for the unknown parameters based on (i) the asymptotic normality of the maximum likelihood estimators (MLEs), and (ii) parametric bootstrap resampling technique. A Monte Carlo simulation study is carried out to examine the performance of these methods of inference. Finally, a numerical example is presented in order to illustrate all the methods of inference developed here.     

Planning step-stress test plans under Type-I censoring for the log-location-scale case

In this paper, we consider a k-level step-stress accelerated life-testing (ALT) experiment with unequal duration steps τ=(τ₁, …, τ _k ). Censoring is allowed only at the change-stress point in the final stage. A general log-location-scale lifetime distribution with mean life which is a linear function of stress, along with a cumulative exposure model, is considered as the working model. Under this model, the determination of the optimal choice of τ for both Weibull and lognormal distributions are addressed using the variance–optimality criterion. Numerical results show that for a general log-location-scale distributions, the optimal k-step-stress ALT model with unequal duration steps reduces just to a 2-level step-stress ALT model.     

PREDICTION FOR EXPONENTIAL LIFETIMES BASED ON STEP-STRESS TESTING

ABSTRACT

This paper presents methods for constructing prediction limits for a step-stress model in accelerated life testing. An exponential life distribution with a mean that is a log-linear function of stress, and a cumulative exposure model are assumed. Two prediction problems are discussed. One concerns the prediction of the life at a design stress, and the other concerns the prediction of a future life during the step-stress testing. Both predictions require the knowledge of some model parameters. When estimates for the model parameters are available, a calibration method based on simulations is proposed for correcting the prediction intervals (regions) obtained by treating the parameter estimates as the true parameter values. Finally, a numerical example is given to illustrate the prediction procedure.     

A GLM approach to step-stress accelerated life testing with interval censoring

In this paper, we present a statistical inference procedure for the step-stress accelerated life testing (SSALT) model with Weibull failure time distribution and interval censoring via the formulation of generalized linear model (GLM). The likelihood function of an interval censored SSALT is in general too complicated to obtain analytical results. However, by transforming the failure time to an exponential distribution and using a binomial random variable for failure counts occurred in inspection intervals, a GLM formulation with a complementary log-log link function can be constructed. The estimations of the regression coefficients used for the Weibull scale parameter are obtained through the iterative weighted least square (IWLS) method, and the shape parameter is updated by a direct maximum likelihood (ML) estimation. The confidence intervals for these parameters are estimated through bootstrapping. The application of the proposed GLM approach is demonstrated by an industrial example.     

Reliability analysis of masked data in adaptive step-stress partially accelerated lifetime tests with progressive removal

By combining the progressive hybrid censoring with the step-stress partially accelerated lifetime test, we propose an adaptive step-stress partially accelerated lifetime test, which allows random changing of the number of step-stress levels according to the pre-fixed censoring number and time points. Thus, the time expenditure and economic cost of the test will be reduced greatly. Based on the Lindley-distributed tampered failure rate (TFR) model with masked system lifetime data, the BFGS method is introduced in the expectation maximization (EM) algorithm to obtain the maximum likelihood estimation (MLE), which overcomes the difficulties of the vague maximization procedure in the M-step. Asymptotic confidence intervals of components' distribution parameters are also investigated according to the missing information principle. As comparison, the Bayesian estimation and the highest probability density (HPD) credible intervals are obtained by using adaptive rejection sampling. Furthermore, the reliability of the system and components are estimated at a specified time under usual and severe operating conditions. Finally, a numerical simulation example is presented to illustrate the performance of our proposed method.     

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.     

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.     

On the simple step-stress model for two-parameter exponential distribution

In this paper, we consider the simple step-stress model for a two-parameter exponential distribution, when both the parameters are unknown and the data are Type-II censored. It is assumed that under two different stress levels, the scale parameter only changes but the location parameter remains unchanged. It is observed that the maximum likelihood estimators do not always exist. We obtain the maximum likelihood estimates of the unknown parameters whenever they exist. We provide the exact conditional distributions of the maximum likelihood estimators of the scale parameters. Since the construction of the exact confidence intervals is very difficult from the conditional distributions, we propose to use the observed Fisher Information matrix for this purpose. We have suggested to use the bootstrap method for constructing confidence intervals. Bayes estimates and associated credible intervals are obtained using the importance sampling technique. Extensive simulations are performed to compare the performances of the different confidence and credible intervals in terms of their coverage percentages and average lengths. The performances of the bootstrap confidence intervals are quite satisfactory even for small sample sizes.     

Optimum Design for Type-I Step-stress Accelerated Life Tests of Two-parameter Weibull Distributions

In this article, we focus on the general k-step step-stress accelerated life tests with Type-I censoring for two-parameter Weibull distributions based on the tampered failure rate (TFR) model. We get the optimum design for the tests under the criterion of the minimization of the asymptotic variance of the maximum likelihood estimate of the pth percentile of the lifetime under the normal operating conditions. Optimum test plans for the simple step-stress accelerated life tests under Type-I censoring are developed for the Weibull distribution and the exponential distribution in particular. Finally, an example is provided to illustrate the proposed design and a sensitivity analysis is conducted to investigate the robustness of the design.     

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 Generalized Type-I Hybrid Censored Data from the Exponential Distribution

In this article, the simple step-stress model is considered based on generalized Type-I hybrid censored data from the exponential distribution. The maximum likelihood estimators (MLEs) of the unknown parameters are derived assuming a cumulative exposure model. We then derive the exact distributions of the MLEs of the parameters using conditional moment generating functions. The Bayesian estimators of the parameters are derived and then compared with the MLEs. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs, Bayesian, and the parametric bootstrap methods. The problem of determining the optimal stress-changing point is discussed and the MLEs of the pth quantile and reliability functions at the use condition are obtained. Finally, Monte Carlo simulation and some numerical results are presented for illustrating all the inferential methods developed here.     

Estimation in step-stress partially accelerated life tests for the Burr type XII distribution using type I censoring

In this paper, step-stress partially accelerated life tests are considered when the lifetime of a product follows a Burr type XII distribution. Based on type I censoring, the maximum likelihood estimates (MLEs) are obtained for the distribution parameters and acceleration factor. In addition, asymptotic variance and covariance matrix of the estimators are given. An iterative procedure is used to obtain the estimators numerically using Mathcad (2001). Furthermore, confidence intervals of the estimators are presented. Simulation results are carried out to study the precision of the MLEs for the parameters involved.     

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.     

Optimal step-stress test under type I progressive group-censoring with random removals

Some traditional life tests result in no or very few failures by the end of test. In such cases, one approach is to do life testing at higher-than-usual stress conditions in order to obtain failures quickly. This paper discusses a k-level step-stress accelerated life test under type I progressive group-censoring with random removals. An exponential failure time distribution with mean life that is a log-linear function of stress and a cumulative exposure model are considered. We derive the maximum likelihood estimators of the model parameters and establish the asymptotic properties of the estimators. We investigate four selection criteria which enable us to obtain the optimum test plans. One is to minimize the asymptotic variance of the maximum likelihood estimator of the logarithm of the mean lifetime at use-condition, and the other three criteria are to maximize the determinant, trace and the smallest eigenvalue of Fisher's information matrix. Some numerical studies are discussed to illustrate the proposed criteria.     

Estimation of the extended Weibull parameters and acceleration factors in the step-stress accelerated life tests under an adaptive progressively hybrid censoring data

ABSTRACT

Based on the tampered failure rate model under the adaptive Type-II progressively hybrid censoring data, we discuss the maximum likelihood estimators of the unknown parameters and acceleration factors in the general step-stress accelerated life tests in this paper. We also construct the exact and unique confidence interval for the extended Weibull shape parameter. In the numerical analysis, we describe the simulation procedures to obtain the adaptive Type-II progressively hybrid censoring data in the step-stress accelerated life tests and present an experimental data to illustrate the performance of the estimators.     

Optimal step-stress testing for progressively Type-I censored data from exponential distribution

In this paper, a k  -step-stress accelerated life-testing is considered with an equal step duration τ$\tau$. For small to moderate sample sizes, a practical modification is made to the model previously considered by Gouno et al. [2004. Optimal step-stress test under progressive Type-I censoring. IEEE Trans. Reliability 53, 383–393] in order to guarantee a feasible k  -step-stress test under progressive Type-I censoring, and the optimal τ$\tau$ is determined under this model. Next, we discuss the determination of optimal τ$\tau$ under the condition that the step-stress test proceeds to the k  -th stress level, and the efficiency of this conditional inference is compared to that of the previous case. In all cases considered, censoring is allowed at each point of stress change (viz., iτ$i\tau$, i=1,2,…,k$i=1,2,\dots ,k$). The determination of optimal τ$\tau$ is discussed under C-optimality, D-optimality, and A-optimality criteria. We investigate in detail the case of progressively Type-I right censored data from an exponential distribution with a single stress variable.