Simple Step-stress Model Research Articles

Bayesian acceptance sampling plan for simple step-stress model

This paper presents a Bayesian acceptance sampling plan for simple step-stress life testing model, when the stress changing time is random, and the experimental units are subject to Type-II censoring. It is assumed that the lifetime distribution of the experimental item is exponential at each stress level with different scale parameters. We first present the Bayes decision function under a specified loss function. Based on the Bayes decision function, the optimal Bayesian sampling plan is determined by minimizing the Bayes risk. We also present some optimal Bayesian sampling plan under specific cases. We further provide order restricted optimal Bayesian sampling plan also. The Bayesian sampling plans presented in this paper are quite beneficial when the experimental items are highly reliable, which is quite common in present days.

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.

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.

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.

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

Exact Inference of a Simple Step-Stress Model with Hybrid Type-II Stress Changing Time

In this article, we consider a simple step-stress model for exponentially distributed lifetime units. As failure rate is lower at the initial stress level, therefore, it is important to pay more attention to the stress changing time. Here, we consider a simple step-stress model where the stress level changes either after a prefixed time or after a prefixed number of failures, whichever occurs later. It ensures a prefixed minimum number of failures at the first stress level and also sets up a control on the expected experimental time. We have obtained the maximum likelihood estimators of the model parameters along with their exact distributions. The monotonicity properties of the maximum likelihood estimators have been established here, and it can be used to construct the exact confidence intervals of the unknown parameters. We provide approximate and bias-corrected accelerated bootstrap confidence intervals of the model parameters. We also define an optimality criteria and based on that obtain an optimal stress changing time for a given sample size. Finally, an extensive simulation study has been performed to asses the performance of the proposed methods and provide the analyses of two data sets for illustrative purpose.

Analysis of Weibull Step-Stress Model In Presence of Competing Risk

In this paper, we mainly consider the inference of a simple step-stress model based on a complete sample, when the stress changes after a prefixed number of failures. It is assumed that there is more than one cause of failure, and the lifetime of the experimental units at each stress level follows Weibull distribution with the same shape parameter and different scale parameters. The distribution function under different stress levels are connected through the generalized Khamis–Higgins model. The maximum likelihood estimates of the model parameters and the associated asymptotic confidence intervals are obtained. Further, we consider the Bayesian inference of the unknown model parameters based on fairly general prior distributions. We have also provided the results for Type-I censored data also. We assess the performances of the estimators through extensive simulation study for complete sample, and the analyses of one complete (simulated) data set and one Type-I censored solar lighting device data set have been performed for illustrative purpose. We propose different classical and Bayesian optimal criteria, and based on them we obtain the optimum stress changing time. Finally, we have indicated how the assumption on the common shape parameter of two competing causes can be relaxed.

Inference for a simple step-stress model based on ordered ranked set sampling

In this paper, we considered the inference problem on simple step-stress accelerated life test data from one-parameter exponential distribution under type-I censored ordered ranked set sample with cumulative exposure model. The Bayesian estimators and credible intervals for the model parameters are developed and compared with the corresponding estimators based on simple random sampling. Two real data sets and numerical simulation evaluations are presented to illustrate all the results developed here. The simulation study indicated that the proposed Bayes estimators and credible intervals based on ordered ranked set sampling performed better than their counterparts using simple random sampling.

Exact Likelihood-Ratio Tests for a Simple Step-Stress Cumulative Exposure Model with Censored Exponential Data

An accelerated life test (ALT) is commonly used in reliability studies of high quality products. A step-stress experiment is one such ALT that is often used in practice. In this paper, we develop exact inference based on likelihood-ratio test for a simple step-stress model with exponential lifetimes under Type-II censoring. We consider both unrestricted and order-restricted maximum likelihood estimators (MLEs) for this purpose. We also derive the exact power function of this test, which we then use for designing an optimal step-stress testing experiment. By using a similar approach, we extend these results to other forms of censoring. Monte Carlo simulations are then carried out to evaluate the performance of the inferential methods developed here. Finally, an illustrative example is presented for demonstrating all the inferential results.

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 inference of a multiple step-stress model

In this manuscript both the classical and Bayesian analyses of a multiple step-stress model have been considered. The lifetime distributions of the experimental units at each stress level follow two-parameter generalized exponential distribution and they are related through the cumulative exposure model assumptions. Recently Abdel-Hamid and AL-Hussaini (2009) provided the classical inference of the model parameters of a simple step-stress model, under the same set of assumptions. In a typical step-stress experiment, it is expected that the lifetime of the experimental units will be shorter at the higher stress level. The main aim of this paper is to develop the order restricted inference of the model parameters of a multiple step-stress model based on both the classical and Bayesian approaches. An extensive simulation study has been performed and one data set has been analyzed for illustrative purposes.

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.

Inference for a simple step-stress model with progressively censored competing risks data from Weibull distribution

ABSTRACTIn reliability analysis, it is common to consider several causes, either mechanical or electrical, those are competing to fail a unit. These causes are called “competing risks.” In this paper, we consider the simple step-stress model with competing risks for failure from Weibull distribution under progressive Type-II censoring. Based on the proportional hazard model, we obtain the maximum likelihood estimates (MLEs) of the unknown parameters. The confidence intervals are derived by using the asymptotic distributions of the MLEs and bootstrap method. For comparison, we obtain the Bayesian estimates and the highest posterior density (HPD) credible intervals based on different prior distributions. Finally, their performance is discussed through simulations.

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.

Order restricted Bayesian inference for exponential simple step-stress model

ABSTRACTA 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.

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.

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

ABSTRACTIn 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.