Simple Step Stress Research Articles

Statistical Inference in the Cumulative Exposure Lognormal Model with Hybrid Censoring

This research aims to analyze data coming from step stress life testing experiments where the stress level is incremented at a preset time to obtain failure data faster. To analyze step stress data, a model that extrapolates the information attained from the accelerated tests to normal conditions needs to be fit to the life test data. We used the Cumulative Exposure Model (CEM) to model simple step stress lognormal life test data where hybrid censoring is present and applied the maximum likelihood estimation method to find the point and interval estimates of the parameters. Bootstrap intervals (bootstrap-t intervals and percentile intervals) were also constructed. We then performed a simulation study to assess the proposed methods of estimation under different hybrid censoring schemes. The Bias and MSE of the maximum likelihood estimators (MLEs) along with the coverage probability and average lengths of the corresponding confidence intervals were investigated. Finally, an illustrative example has been used to demonstrate the application of the methods discussed in this paper.

E-Bayesian and H-Bayesian Inferences for a Simple Step-Stress Model with Competing Failure Model under Progressively Type-II Censoring

In this paper, we discuss the statistical analysis of a simple step-stress accelerated competing failure model under progressively Type-II censoring. It is assumed that there is more than one cause of failure, and the lifetime of the experimental units at each stress level follows exponential distribution. The distribution functions under different stress levels are connected through the cumulative exposure model. The maximum likelihood, Bayesian, Expected Bayesian, and Hierarchical Bayesian estimations of the model parameters are derived based on the different loss function. Based on Monte Carlo Simulations. We also get the average length and the coverage probability of the 95% confidence intervals and highest posterior density credible intervals of the parameters. From the numerical studies, it can be seen that the proposed Expected Bayesian estimations and Hierarchical Bayesian estimations have better performance in terms of the average estimates and mean squared errors, respectively. Finally, the methods of statistical inference discussed here are illustrated with a numerical example.

Expected Bayesian estimation for exponential model based on simple step stress with Type-I hybrid censored data.

The procedure of selecting the values of hyper-parameters for prior distributions in Bayesian estimate has produced many problems and has drawn the attention of many authors, therefore the expected Bayesian (E-Bayesian) estimation method to overcome these problems. These approaches are used based on the step-stress acceleration model under the Exponential Type-I hybrid censored data in this study. The values of the distribution parameters are derived. To compare the E-Bayesian estimates to the other estimates, a comparative study was conducted using the simulation research. Four different loss functions are used to generate the Bayesian and E-Bayesian estimators. In addition, three alternative hyper-parameter distributions were used in E-Bayesian estimation. Finally, a real-world data example is examined for demonstration and comparative purposes.

Statistical Inference for a Simple Step Stress Model with Competing Risks Based on Generalized Type-I Hybrid Censoring

Abstract This paper investigates a simple step-stress accelerated lifetime test (SSALT) model for the inferential analysis of exponential competing risks data. A generalized type-I hybrid censoring scheme is employed to improve the efficiency and controllability of the test. Firstly, the MLEs for parameters are established based on the cumulative exposure model (CEM). Then the conditional moment generating function (MGF) for unknown parameters is set up using conditional expectation and multiple integral techniques. Thirdly, confidence intervals (CIs) are constructed by the exact MGF-based method, the approximate normality-based method, and the bias-corrected and accelerated (BCa) percentile bootstrap method. Finally, we present simulation studies and an illustrative example to compare the performances of different methods.

Designing Bayesian sampling plans for simple step-stress of accelerated life test on censored data

ABSTRACT This paper studies the problem about how to design a Bayesian sampling plan (BSP) for two exponential distributions linked by the cumulative exposure model through a simple step-stress accelerated life test (ALT). Such a Bayesian sampling plan through the ALT by a simple step-stress procedure is called BSPA. The BSPA with Type-II censoring in a general loss function is derived. Given joint gamma and uniform prior distributions, an explicit Bayes decision function under a certain loss function is derived. Illustrative examples are given to demonstrate how to find the Bayes decision function. A Monte Carlo simulation study is performed for searching the optimal BSPA. Comparison between the proposed BSPA and the conventional BSP is carried out to study the performance of BSPA. The numerical results indicate that the risk reduction of BSPA by applying the accelerated procedure is more significant if the experimental time cost is more expensive.

Simple Step-Stress Model with Progressively Censored Competing Risks Data from Extension of Weibull Distribution

Simple Step-Stress Model with Progressively Censored Competing Risks Data from Extension of Weibull Distribution

Statistical Inference For A Simple Step–Stress Model With Type–II Hybrid Censored Data From The Kumaraswamy Weibull Distribution

In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress factors on the lifetimes of experimental units. In this paper, a step-stress model is considered in which the life-testing experiment gets terminated either at a pre-fixed time (say, Tm+1) or at a random time ensuring at least a specified number of failures (Say, y out of n). Under this model in which the data obtained are Type-II hybrid censored, the Kumaraswamy Weibull distribution is used for the underlying lifetimes. The maximum Likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model are derived. The confidence intervals of the parameters are also obtained. The hazard rate and reliability functions are estimated at usual conditions of stress. Monte Carlo simulation is carried out to investigate the precision of the maximum likelihood estimates. An application using real data is used to indicate the properties of the maximum likelihood estimators.

E‐Bayesian estimation and associated properties of simple step–stress model for exponential distribution based on type‐II censoring

AbstractIn this paper, expected Bayesian (E‐Bayesian) estimation for the simple step–stress model based on type‐II censoring scheme is considered. The case of exponential distribution for the underlying lifetimes is considered assuming a cumulative exposure model. The E‐Bayesian estimation is discussed by considering three different prior distributions for the hyperparameters. The E‐Bayesian estimators as well as the corresponding E‐posterior risks are obtained by using squared error and linear‐exponential (LINEX) loss functions. Some properties of the E‐Bayesian estimators are also derived. We conduct a simulation study to compare the various estimators and a simulated and real data sets are analyzed to show the applicability of the different estimators. The numerical results show that the E‐Bayesian estimators perform better than the classical and Bayesian estimators.

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.

On Bayesian estimation of step stress accelerated life testing for exponentiated Lomax distribution based on censored samples

In reliability analysis and life-testing experiments, the researcher is often interested in the effects of changing stress factors such as “temperature”, “voltage” and “load” on the lifetimes of the units. Step-stress (SS) test, which is a special class from the well-known accelerated life-tests, allows the experimenter to increase the stress levels at some constant times to obtain information on the unknown parameters of the life models more speedily than under usual operating conditions. In this paper, a simple SS model from the exponentiated Lomax (ExpLx) distribution when there is time limitation on the duration of the experiment is considered. Bayesian estimates of the parameters assuming a cumulative exposure model with lifetimes being ExpLx distribution are resultant using Markov chain Monte Carlo (M.C.M.C) procedures. Also, the credible intervals and predicted values of the scale parameter, reliability and hazard are derived. Finally, the numerical study and real data are presented to illustrate the proposed study

Estimation for a simple step–stress model with type–II hybrid censored data from the exponentiated Rayleigh distribution

Estimation for a simple step–stress model with type–II hybrid censored data from the exponentiated Rayleigh distribution

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.

Order Restricted Bayesian Analysis of a Simple Step Stress Model

In this article we consider a simple step stress set up under the cumulative exposure model assumption. At each stress level the lifetime distribution of the experimental units are assumed to follow the generalized exponential distribution. We provide the order restricted Bayesian inference of the model parameters by considering the fact that the expected lifetime of the experimental units are larger in lower stress level. Analysis and the related results are extended to different censoring schemes also. The Bayes estimates and the associated credible intervals of the unknown parameters are constructed using importance sampling technique. We perform extensive simulation experiments both for the complete and censored samples to see the performances of the proposed estimators. We analyze two simulated and one real data sets for illustrative purposes. An optimal value of the stress changing time is obtained by minimizing the total posterior coefficient of variations of the unknown parameters.

Point and interval estimation for a simple step-stress model with Type-I censored data from geometric distribution

The estimation problem of expected time to failure of units is studied in a discrete set up. A simple stepstress accelerated life testing is considered with a Type-I censored sample from geometric distribution that is a commonly used distribution to model the lifetime of a device in discrete case. Maximum likelihood estimators as well as the associated distributions are derived. Exact, approximate and bootstrap approaches construct confidence intervals that are compared via a simulation study. Optimal confidence intervals are suggested in view of the expected width and coverage probability criteria. An illustrative example is also presented to explain the results of the paper. Finally, some conclusions are stated.

Estimation for Birnbaum–Saunders Distribution in Simple Step Stress–accelerated Life Test with Type-II Censoring

This paper develops the Bayesian estimation for the Birnbaum–Saunders distribution based on Type-II censoring in the simple step stress–accelerated life test with power law accelerated form. Maximum likelihood estimates are obtained and Gibbs sampling procedure is used to get the Bayesian estimates for shape parameter of Birnbaum–Saunders distribution and parameters of power law–accelerated model. Asymptotic normality method and Markov Chain Monte Carlo method are employed to construct the corresponding confidence interval and highest posterior density interval at different confidence level, respectively. At last, the results are compared by using Monte Carlo simulations, and a numerical example is analyzed for illustration.

Graphical Representation of Optimal Time for a Step-Stress Accelerated Life Test Design Using Frechet Distribution

The article provides an approach of getting optimal time through graph for Simple step stress accelerated test of inverse weibull distribution. In this we estimate parameters using log linear relationship by maximum likelihood method. Along with this, asymptotic variance and covariance matrix of the estimators are given. Comparison between expected and observed Fisher Information matrix is also shown. Furthermore, confidence interval coverage of the estimators is also presented for checking the precession of estimator. This approach is illustrated with an example using software.

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.

Prediction of Times to Failure of Censored Items for a Simple Step-Stress Model with Regular and Progressive Type I Censoring from the Exponential Distribution

The problem of predicting times to failure of units from the Exponential Distribution which are censored under a simple step-stress model is considered in this article. We discuss two types of censoring—regular and progressive Type I—and two kinds of predictors—the maximum likelihood predictors (MLP) and the conditional median predictors (CMP) for each type of censoring. Numerical examples are used to illustrate the prediction methods. Using simulation studies, mean squared prediction error (MSPE) and prediction intervals are generated for these examples. MLP and the CMP are then compared with respect to MSPE and the prediction interval.

Statistical Inference for a Simple Step-Stress Model Based on Censored Data from the Kumaraswamy Weibull Distribution

The step-stress accelerated life tests allow increasing the stress levels on test units at fixed time during the experiment. In this paper, accelerated life tests are considered when lifetime of a product follows a Kumaraswamy Weibull distribution. The shape parameter is assumed to be a log linear function of the stress and a cumulative exposure model holds. Based on Type II and Type I censoring, the maximum likelihood estimates are obtained for the unknown parameters. The reliability and hazard rate functions are estimated at usual conditions of stress. In addition, confidence intervals of the estimators are constructed. Optimum test plans are obtained to minimize the generalized asymptotic variance of the maximum likelihood estimators. Monte Carlo simulation is carried out to investigate the precision of the maximum likelihood estimates. An application using real data is used to indicate the properties of the maximum likelihood estimators.

Bayes Estimation for a Simple Step-stress Model with Type-I Censored Data from the Geometric Distribution

This paper focuses on a Bayes inference model for a simple stepstress life test using Type-I censored sample in a discrete set-up. Assuming the failure times at each stress level are geometrically distributed, the Bayes estimation problem of the parameters of interest is investigated in the both of point and interval approaches. To derive the Bayesian point estimators, some various balanced loss functions are used. Furthermore, a simulation study and sensitivity analysis is performed to carry out the performance of the results of the paper. An example is also presented to illustrate the proposed procedure. Finally, some conclusions are stated.