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.     

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.     

On Parameter Inference for Step-Stress Accelerated Life Test with Geometric Distribution

In this article, we present the parameter inference in step-stress accelerated life tests under the tampered failure rate model with geometric distribution. We deal with Type-II censoring scheme involved in experimental data, and provide the maximum likelihood estimate and confidence interval of the parameters of interest. With the help of the Monte-Carlo simulation technique, a comparison of precision of the confidence limits is demonstrated for our method, the Bootstrap method, and the large-sample based procedure. The application of two industrial real datasets shows the proposed method efficiency and feasibility.     

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.     

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.     

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.     

Inference for the Birnbaum–Saunders Lifetime Regression Model with Applications

The Birnbaum–Saunders distribution is a positively skewed distribution that is frequently used for analyzing lifetime data. Regression analysis is widely used in this context when some covariates are involved in the life-test. In this article, we discuss the maximum likelihood estimation of the model parameters and associated inference. We discuss the likelihood-ratio tests for some hypotheses of interest as well as some interval estimation methods. A Monte Carlo simulation study is then carried out to examine the performance of the proposed estimators and the interval estimation methods. Finally, some numerical data analyses are done for illustrating all the inferential methods developed here.     

Maximum likelihood estimation of the parameters of a multiple step-stress model from the Birnbaum-Saunders distribution under time-constraint: A comparative study

The cumulative exposure model (CEM) is a commonly used statistical model utilized to analyze data from a step-stress accelerated life testing which is a special class of accelerated life testing (ALT). In practice, researchers conduct ALT to: (1) determine the effects of extreme levels of stress factors (e.g., temperature) on the life distribution, and (2) to gain information on the parameters of the life distribution more rapidly than under normal operating (or environmental) conditions. In literature, researchers assume that the CEM is from well-known distributions, such as the Weibull family. This study, on the other hand, considers a p-step-stress model with q stress factors from the two-parameter Birnbaum-Saunders distribution when there is a time constraint on the duration of the experiment. In this comparison paper, we consider different frameworks to numerically compute the point estimation for the unknown parameters of the CEM using the maximum likelihood theory. Each framework implements at least one optimization method; therefore, numerical examples and extensive Monte Carlo simulations are considered to compare and numerically examine the performance of the considered estimation frameworks.     

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.     

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.     

Inference for Weibull distribution based on progressively Type-II hybrid censored data

Progressive Type-II hybrid censoring is a mixture of progressive Type-II and hybrid censoring schemes. In this paper, we discuss the statistical inference on Weibull parameters when the observed data are progressively Type-II hybrid censored. We derive the maximum likelihood estimators (MLEs) and the approximate maximum likelihood estimators (AMLEs) of the Weibull parameters. We then use the asymptotic distributions of the maximum likelihood estimators to construct approximate confidence intervals. Bayes estimates and the corresponding highest posterior density credible intervals of the unknown parameters are obtained under suitable priors on the unknown parameters and also by using the Gibbs sampling procedure. Monte Carlo simulations are then performed for comparing the confidence intervals based on all those different methods. Finally, one data set is analyzed for illustrative purposes.     

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.     

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.     

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.     

Parameter estimation of Lindley step stress model with independent competing risk under type 1 censoring

Abstract

In literature, Lindley distribution is considered as an alternative to exponential distribution to fit lifetime data. In the present work, a Lindley step-stress model with independent causes of failure is proposed. An algorithm to generate random samples from the proposed model under type 1 censoring scheme is developed. Point and interval estimation of the model parameters is carried out using maximum likelihood method and percentile bootstrap approach. To understand the effectiveness of the resulting estimates, numerical illustration is provided based on simulated and real-life data sets.     

Standard errors for the parameters of graphical Gaussian models

The comparison of an estimated parameter to its standard error, the Wald test, is a well known procedure of classical statistics. Here we discuss its application to graphical Gaussian model selection. First we derive the Fisher information matrix and its inverse about the parameters of any graphical Gaussian model. Both the covariance matrix and its inverse are considered and a comparative analysis of the asymptotic behaviour of their maximum likelihood estimators (m.l.e.s) is carried out. Then we give an example of model selection based on the standard errors. The method is shown to produce almost identical inference to likelihood ratio methods in the example considered.     

Bayesian analysis for step-stress accelerated life testing using weibull proportional hazard model

In this paper, we present a Bayesian analysis for the Weibull proportional hazard (PH) model used in step-stress accelerated life testings. The key mathematical and graphical difference between the Weibull cumulative exposure (CE) model and the PH model is illustrated. Compared with the CE model, the PH model provides more flexibility in fitting step-stress testing data and has the attractive mathematical properties of being desirable in the Bayesian framework. A Markov chain Monte Carlo algorithm with adaptive rejection sampling technique is used for posterior inference. We demonstrate the performance of this method on both simulated and real datasets.     

Inference for Log-Gamma Distribution Based on Progressively Type-II Censored Data

We discuss the maximum likelihood estimates (MLEs) of the parameters of the log-gamma distribution based on progressively Type-II censored samples. We use the profile likelihood approach to tackle the problem of the estimation of the shape parameter κ. We derive approximate maximum likelihood estimators of the parameters μ and σ and use them as initial values in the determination of the MLEs through the Newton–Raphson method. Next, we discuss the EM algorithm and propose a modified EM algorithm for the determination of the MLEs. A simulation study is conducted to evaluate the bias and mean square error of these estimators and examine their behavior as the progressive censoring scheme and the shape parameter vary. We also discuss the interval estimation of the parameters μ and σ and show that the intervals based on the asymptotic normality of MLEs have very poor probability coverages for small values of m. Finally, we present two examples to illustrate all the methods of inference discussed in this paper.     

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.     

Parameter estimation and prediction of order statistics for the Burr Type XII distribution with Type II censoring

This article deals with the statistical inference and prediction on Burr Type XII parameters based on Type II censored sample. It is observed that the maximum likelihood estimators (MLEs) cannot be obtained in closed form. We use the expectation-maximization algorithm to compute the MLEs. We also obtain the Bayes estimators under symmetric and asymmetric loss functions such as squared error and Linex By applying Lindley's approximation and Markov chain Monte Carlo (MCMC) technique. Further, MCMC samples are used to calculate the highest posterior density credible intervals. Monte Carlo simulation study and two real-life data-sets are presented to illustrate all of the methods developed here. Furthermore, we obtain a prediction of future order statistics based on the observed ordered because of its important application in different fields such as medical and engineering sciences. A numerical example carried out to illustrate the procedures obtained for prediction of future order statistics.