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.     

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.     

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.     

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.     

Multiple-Steps Step-Stress Accelerated Degradation Modeling Based on Wiener and Gamma Processes

During the step-stress accelerated degradation test (SSADT) experiment, the operator usually elevates the stress level at a predetermined time-point for all tested products that had not failed. This time-point is determined by the experience of the operator and the test is carried on until the performance degradation value of the product crosses the threshold value. In fact, this mode only works when a lot of products have been used in the experiment. But in the SSADT experiment, the number of products is relatively few, and so the test control to the products should be done more carefully. Considering the differences in products, we think the time-point of elevating stress level varies randomly from product-to-product. We consider a situation in which when the degradation value crosses a pre-specified value, the stress level is elevated. Under the circumstances, the time at which the degradation path crosses the pre-specified value is uncertain, and so we may regard it as a random variable. We discuss multiple-steps step-stress accelerated degradation models based on Wiener and gamma processes, respectively, and we apply the Bayesian Markov chain Monte Carlo (MCMC) method for such analytically intractable models to obtain the maximum likelihood estimates (MLEs) efficiently and present some computational results obtained from our implementation.     

Rank Regression Analysis of Multivariate Failure Time Data Based on Marginal Linear Models

Abstract.  Multivariate failure time data arises when each study subject can potentially ex-perience several types of failures or recurrences of a certain phenomenon, or when failure times are sampled in clusters. We formulate the marginal distributions of such multivariate data with semiparametric accelerated failure time models (i.e. linear regression models for log-transformed failure times with arbitrary error distributions) while leaving the dependence structures for related failure times completely unspecified. We develop rank-based monotone estimating functions for the regression parameters of these marginal models based on right-censored observations. The estimating equations can be easily solved via linear programming. The resultant estimators are consistent and asymptotically normal. The limiting covariance matrices can be readily estimated by a novel resampling approach, which does not involve non-parametric density estimation or evaluation of numerical derivatives. The proposed estimators represent consistent roots to the potentially non-monotone estimating equations based on weighted log-rank statistics. Simulation studies show that the new inference procedures perform well in small samples. Illustrations with real medical data are provided.     

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.     

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.     

Choice of Parametric Accelerated Life and Proportional Hazards Models for Survival Data: Asymptotic Results

We discuss the impact of misspecifying fully parametric proportional hazards and accelerated life models. For the uncensored case, misspecified accelerated life models give asymptotically unbiased estimates of covariate effect, but the shape and scale parameters depend on the misspecification. The covariate, shape and scale parameters differ in the censored case. Parametric proportional hazards models do not have a sound justification for general use: estimates from misspecified models can be very biased, and misleading results for the shape of the hazard function can arise. Misspecified survival functions are more biased at the extremes than the centre. Asymptotic and first order results are compared. If a model is misspecified, the size of Wald tests will be underestimated. Use of the sandwich estimator of standard error gives tests of the correct size, but misspecification leads to a loss of power. Accelerated life models are more robust to misspecification because of their log-linear form. In preliminary data analysis, practitioners should investigate proportional hazards and accelerated life models; software is readily available for several such models.     

Hyper Inverse Wishart Distribution for Non-decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models

While conjugate Bayesian inference in decomposable Gaussian graphical models is largely solved, the non-decomposable case still poses difficulties concerned with the specification of suitable priors and the evaluation of normalizing constants. In this paper we derive the DY-conjugate prior ( Diaconis & Ylvisaker, 1979 ) for non-decomposable models and show that it can be regarded as a generalization to an arbitrary graph G of the hyper inverse Wishart distribution ( Dawid & Lauritzen, 1993 ). In particular, if G is an incomplete prime graph it constitutes a non-trivial generalization of the inverse Wishart distribution. Inference based on marginal likelihood requires the evaluation of a normalizing constant and we propose an importance sampling algorithm for its computation. Examples of structural learning involving non-decomposable models are given. In order to deal efficiently with the set of all positive definite matrices with non-decomposable zero-pattern we introduce the operation of triangular completion of an incomplete triangular matrix. Such a device turns out to be extremely useful both in the proof of theoretical results and in the implementation of the Monte Carlo procedure.     

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.     

Optimal bivariate step-stress accelerated life test for Type-I hybrid censored data

This paper presents a step-stress accelerated life test for two stress variables to obtain optimal hold times under a Type-I hybrid censoring scheme. An exponentially distributed life and a cumulative exposure model are assumed. The maximum-likelihood estimates are given, from which the asymptotic variance and the Fisher information matrix are obtained. The optimal test plan is determined for each combination of stress levels by minimizing the asymptotic variance of reliability estimate at a typical operating condition. Finally, simulation results are discussed to illustrate the proposed criteria. Simulation results show that the proposed optimum plan is robust, and the initial estimates have a small effect on optimal values.     

Failure Inference From a Marker Process Based on a Bivariate Wiener Model

Many models have been proposed that relate failure times and stochastic time-varying covariates. In some of these models, failure occurs when a particular observable marker crosses a threshold level. We are interested in the more difficult, and often more realistic, situation where failure is not related deterministically to an observable marker. In this case, joint models for marker evolution and failure tend to lead to complicated calculations for characteristics such as the marginal distribution of failure time or the joint distribution of failure time and marker value at failure. This paper presents a model based on a bivariate Wiener process in which one component represents the marker and the second, which is latent (unobservable), determines the failure time. In particular, failure occurs when the latent component crosses a threshold level. The model yields reasonably simple expressions for the characteristics mentioned above and is easy to fit to commonly occurring data that involve the marker value at the censoring time for surviving cases and the marker value and failure time for failing cases. Parametric and predictive inference are discussed, as well as model checking. An extension of the model permits the construction of a composite marker from several candidate markers that may be available. The methodology is demonstrated by a simulated example and a case application.     

Statistical Inference Based on Progressively Type II Censored Data from Weibull Model

In this article, we consider the problem of estimating the shape and scale parameters and predicting the unobserved removed data based on a progressive type II censored sample from the Weibull distribution. Maximum likelihood and Bayesian approaches are used to estimate the scale and shape parameters. The sampling-based method is used to draw Monte Carlo (MC) samples and it has been used to estimate the model parameters and also to predict the removed units in multiple stages of the censored sample. Two real datasets are presented and analyzed for illustrative purposes and Monte carlo simulations are performed to study the behavior of the proposed methods.     

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.     

On constant stress accelerated life tests terminated by Type II censoring at one of the stress levels

This paper considers constant stress accelerated life tests terminated by a Type II censoring regime at one of the stress levels. We consider a model based on Weibull distributions with constant shape and a log-linear link between scale and the stress factor. We obtain expectations associated with the likelihood function, and use these to obtain asymptotically valid variances and correlations for maximum likelihood estimates of model parameters. We illustrate their calculation, and assess agreement with observed counterparts for finite samples in simulation experiments. We then use moments to compare the information obtained from variants of the design, and show that, with an appropriate allocation of items to stress levels, the design yields better estimates of model parameters and related quantities than a single stress experiment.     

Reliability statistical analysis about products of Birnbaum–Saunders fatigue life distribution for life test and progressive stress accelerated life test

Birnbaum–Saunders fatigue life distribution is an important failure model in the probability physical methods. It is more suitable for describing the life rules of fatigue failure products than common life distributions such as Weibull distribution and lognormal distribution. Besides, it is mainly applied to analytical research about fatigue failure and degradation failure of electronic product performance. The characteristic properties such as numerical characteristics and image features of density function and failure rate function are studied for generalized BS fatigue life distribution GBS(α, β, m) in this paper. Then the point estimates and approximate interval estimates of parameters are proposed for generalized BS fatigue life distribution GBS(α, β, m), and the precision of estimates are investigated by Monte Carlo simulations. Finally, when the scale parameter satisfies inverse power law model, the failure distribution model is given for the products of two-parameter BS fatigue life distribution BS(α, β) under progressive stress accelerated life test according to the time conversion idea of famous Nelson assumption, and then the points estimates of parameters are given.     

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.     

Inference on progressive-stress model for the exponentiated exponential distribution under type-II progressive hybrid censoring

In this paper, progressive-stress accelerated life tests are applied when the lifetime of a product under design stress follows the exponentiated distribution [G(x)]^α. The baseline distribution, G(x), follows a general class of distributions which includes, among others, Weibull, compound Weibull, power function, Pareto, Gompertz, compound Gompertz, normal and logistic distributions. The scale parameter of G(x) satisfies the inverse power law and the cumulative exposure model holds for the effect of changing stress. A special case for an exponentiated exponential distribution has been discussed. Using type-II progressive hybrid censoring and MCMC algorithm, Bayes estimates of the unknown parameters based on symmetric and asymmetric loss functions are obtained and compared with the maximum likelihood estimates. Normal approximation and bootstrap confidence intervals for the unknown parameters are obtained and compared via a simulation study.     

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.