A general class of cumulative damage models for materials failure

A generalized cumulative damage approach is presented which yields a large family of accelerated test inverse Gaussian-type models for strength of materials that incorporate the size effect as the acceleration variable. Previous models are generalized here in three aspects: the cumulative damage model is more general and can include damage functions other than the additive and multiplicative damages; the strength reduction function due to initial damage existing in the material is taken to be a very general function; and the initial damage process is a more general stochastic process that includes those previously assumed as special cases. The approach taken here is therefore the most general cumulative damage model obtained to date and yields a large number of potentially more useful accelerated test models for material strength. Estimation of model parameters by maximum likelihood methods is discussed, and two examples using real tensile strength data for carbon micro-composites and single carbon fibers are presented, illustrating the improvement of the new approach over previous models.     

Reliability computation under dynamic stress–strength modeling with cumulative stress and strength degradation

Reliability function is defined under suitable assumptions for dynamic stress–strength scenarios where strength degrades and stress accumulates over time. Methods for numerical evaluation of reliability are suggested under deterministic strength degradation and cumulative damage due to shocks arriving according to a point process, in particular a Poisson process, using simulation method and inversion theorem. These methods are specifically useful in the scenarios where damage distributions do not possess closure property under convolution. The method is also extended for non-identical, dependent damage distributions as well as for random strength degradation. Results from inversion method is compared with known approximate methods and also verified by simulation. As it turns out, the simulation method seems to have an edge in terms of computational burden and has much wider domain of applicability.     

Time Series Modeling of Degradation Due to Outdoor Weathering

Environmental variables have an important effect on the reliability of many products such as coatings and polymeric composites. Long-term prediction of the performance or service life of such products must take into account the probabilistic/stochastic nature of the outdoor weather. In this article, we propose a time series modeling procedure to model the time series data of daily accumulated degradation. Daily accumulated degradation is the total amount of degradation accrued within one day and can be obtained by using a degradation rate model for the product and the weather data. The fitted model of the time series can then be used to estimate the future distribution of cumulative degradation over a period of time, and to compute reliability measures such as the probability of failure. The modeling technique and estimation method are illustrated using the degradation of a solar reflector material. We also provide a method to construct approximate confidence intervals for the probability of failure.     

On the assessment of tolerance limits under inverse gaussian distribution

Inverse Gaussian distribution has been used in a wide range of applications in modeling duration and failure phenomena. In these applications, one-sided lower tolerance limits are employed, for instance, for designing safety limits of medical devices. Tang and Chang (1994) proposed lowersided tolerance limits via Bonferroni inequality when parameters in the inverse Gaussian distribution are unknown. However, their simulation results showed conservative coverage probabilities, and consequently larger interval width. In their paper, they also proposed an alternative to construct lesser conservative limits. But simulation results yielded unsatisfactory coverage probabilities in many cases. In this article, the exact lower-sided tolerance limit is proposed. The proposed limit has a similar form to that of the confidence interval for mean under inverse Gaussian. The comparison between the proposed limit and Tang and Chang's method is compared via extensive Monte Carlo simulations. Simulation results suggest that the proposed limit is superior to Tang and Chang's method in terms of narrower interval width and approximate to nominal level of coverage probability. Similar argument can be applied to the formulation of two-sided tolerance limits. A summary and conclusion of the proposed limits is included.     

Designing and analyzing a mixture experiment to optimize the mixing proportions of polyvinyl chloride composites

Polyvinyl chloride (PVC) products are typically complex composites, whose quality characteristics vary widely depending on the types and proportions of their components, as well as other processing factors. It is often required to optimize PVC production for specific applications at the highest cost efficiency. This study describes the design and analysis of a statistical experiment to investigate the effects of different parameters over the mechanical properties of PVC intended for use in electrical wire insulation. Four commonly used mixture components, namely, virgin PVC, recycled PVC, calcium carbonate, and a plasticizer, and two process variables, type of plasticizer and filler particle size, were examined. Statistical tools were utilized to analyze and optimize the mixture while simultaneously finding the proper process parameters. The mix was optimized to achieve required strength and ductility, as per ASTM D6096 while minimizing cost. The paper demonstrates how statistical models can help tailor complex polymeric composites in the presence of variations created by process variables.     

Estimation of reliability with cumulative stress and strength degradation

In many real-life scenarios, system reliability depends on dynamic stress–strength interference where strength degrades and stress accumulates concurrently over time. In this paper, we consider the problem of estimating reliability of a system under deterministic strength degradation and cumulative damage due to shocks arriving according to a point process. Maximum likelihood estimation under two different sampling plans has been considered. Large sample properties in general are discussed. The method is illustrated through simulation and real-life data analysis.     

Residual Life Estimation Based on a Generalized Wiener Process with Skew-normal Random Effects

For reliability-critical and expensive products, it is necessary to estimate their residual lives based on available information, such as the degradation data, so that proper maintenance actions can be arranged to reduce or even avoid the occurrence of failures. In this work, by assuming that the product-to-product variability of the degradation is characterized by a skew-normal distribution, a generalized Wiener process-based degradation model is developed. Following that, the issue of residual life (RL) estimation of the target product is addressed in detail. The proposed degradation model provides greater flexibility to capture a variety of degradation processes, since several commonly used Wiener process-based degradation models can be seen as special cases. Through the EM algorism, the population-based degradation information is used to estimate the parameters of the model. Whenever new degradation measurement information of the target product becomes available, the degradation model is first updated based on the Bayesian method. In this way, the RL of the target product can be estimated in an adaptive manner. Finally, the developed methodology is demonstrated by a simulation study.     

Small sample characteristics of generalized estimating equations

The aim of this study was to investigate the Type I error rate of hypothesis testing based on generalized estimating equations (GEE) for data characteristic of periodontal clinical trials. The data in these studies consist of a large number of binary responses from each subject and a small number of subjects (Haffajee et al. (1983), Goodson (1986), Jenkins et al. (1988)) Computer simulations were employed to investigate GEE based both on an empirical estimate of the variance-covariance matrix and a model-based estimate. Results from this investigation indicate that hypothesis testing based on GEE resulted in inappropriate Type I error rates when small samples are employed. Only an increase in the number of subjects to the point where it matched the number of observations per subject resulted in appropriate Type I error rates     

Modelling Accelerated Degradation Data Using Wiener Diffusion With A Time Scale Transformation

Engineering degradation tests allow industry to assess the potential life span of long-life products that do not fail readily under accelerated conditions in life tests. A general statistical model is presented here for performance degradation of an item of equipment. The degradation process in the model is taken to be a Wiener diffusion process with a time scale transformation. The model incorporates Arrhenius extrapolation for high stress testing. The lifetime of an item is defined as the time until performance deteriorates to a specified failure threshold. The model can be used to predict the lifetime of an item or the extent of degradation of an item at a specified future time. Inference methods for the model parameters, based on accelerated degradation test data, are presented. The model and inference methods are illustrated with a case application involving self-regulating heating cables. The paper also discusses a number of practical issues encountered in applications. This revised version was published online in July 2006 with corrections to the Cover Date.     

Tolerance limits under Poisson regression based on small-sample asymptotic methodology

This article explores the calculation of tolerance limits for the Poisson regression model based on the profile likelihood methodology and small-sample asymptotic corrections to improve the coverage probability performance. The data consist of n counts, where the mean or expected rate depends upon covariates via the log regression function. This article evaluated upper tolerance limits as a function of covariates. The upper tolerance limits are obtained from upper confidence limits of the mean. To compute upper confidence limits the following methodologies were considered: likelihood based asymptotic methods, small-sample asymptotic methods to improve the likelihood based methodology, and the delta method. Two applications are discussed: one application relating to defects in semiconductor wafers due to plasma etching and the other examining the number of surface faults in upper seams of coal mines. All three methodologies are illustrated for the two applications.     

Residual life estimation based on nonlinear-multivariate Wiener processes

For some operable products with critical reliability constraints, it is important to estimate accurately their residual lives so that maintenance actions can be arranged suitably and efficiently. In the literature, most publications have dealt with this issue by only considering one-dimensional degradation data. However, this may be not reasonable in situations wherein a product may have two or more performance characteristics (PCs). In such situations, multi-dimensional degradation data should be taken into account. Here, for the target product with multivariate PCs, methods of residual life (RL) estimation are developed. This is done with the assumption that the degradation of PCs over time is governed by a multivariate Wiener process with nonlinear drifts. Both the population-based degradation information and the degradation history of the target product up-to-date are combined to estimate the RL of the product. Specifically, the population-based degradation information is first used to obtain the estimates of the unknown parameters of the model through the EM algorithm. Then, the degradation history of the target product is adopted to update the degradation model, based on which the RL is estimated accordingly. To illustrate the validity and the usefulness of the proposed method, a numerical example about fatigue cracks is finally presented and analysed.     

Statistical Analysis of Linear Degradation and Failure Time Data with Multiple Failure Modes

The paper considers linear degradation and failure time models with multiple failure modes. Dependence of traumatic failure intensities on the degradation level are included into the models. Estimators of traumatic event cumulative intensities, and of various reliability characteristics are proposed. Prediction of residual reliability characteristics given a degradation value at a given moment is discussed. Non-parametric, semiparametric and parametric estimation methods are given. Theorems on simultaneous asymptotic distribution of random functions characterising degradation and intensities of traumatic events are proposed. Asymptotic properties of unconditional and residual reliability characteristics estimators are given. Real tire wear and failure time data are analysed.     

Real-time reliability evaluation of two-phase Wiener degradation process

Reliability modeling and evaluation for the two-phase Wiener degradation process are studied. For many devices, the degradation rates could possibly increase or decrease in a non smooth manner at some point in time due to the change of degradation mechanism. A two-phase Wiener degradation process with an unobserved change point is used to model the degradation process. And we assume that the change point varies randomly from device to device. Furthermore, we integrate historical data and up-to-date observation data to improve the degradation modeling and evaluation based on Bayesian method. The change point between the two phases was obtained based on the Akaike information criterion (AIC) and the criterion of the residual sum of squares. Finally, a real example of liquid coupling devices (LCDs) and a numeric example are discussed to demonstrate the effectiveness of the proposed method. The results show that the proposed method is effective and efficient.     

Residual life estimation under time-varying conditions based on a Wiener process

Residual life (RL) estimation plays an important role in prognostics and health management. In operating conditions, components usually experience stresses continuously varying over time, which have an impact on the degradation processes. This paper investigates a Wiener process model to track and predict the RL under time-varying conditions. The item-to-item variation is captured by the drift parameter and the degradation characteristic of the whole population is described by the diffusion parameter. The bootstrap method and Bayesian theorem are employed to estimate and update the distribution parameters of ‘a’ and ‘b’, which are the coefficients of the linear drifting process in the degradation model. Once new degradation information becomes available, the RL distributions considering the future operating condition are derived. The proposed method is tested on Lithium-ion battery devices under three levels of charging/discharging rates. The results are further validated by a simulation method.     

A new class of survival regression models with heavy-tailed errors: robustness and diagnostics

Birnbaum-Saunders models have largely been applied in material fatigue studies and reliability analyses to relate the total time until failure with some type of cumulative damage. In many problems related to the medical field, such as chronic cardiac diseases and different types of cancer, a cumulative damage caused by several risk factors might cause some degradation that leads to a fatigue process. In these cases, BS models can be suitable for describing the propagation lifetime. However, since the cumulative damage is assumed to be normally distributed in the BS distribution, the parameter estimates from this model can be sensitive to outlying observations. In order to attenuate this influence, we present in this paper BS models, in which a Student-t distribution is assumed to explain the cumulative damage. In particular, we show that the maximum likelihood estimates of the Student-t log-BS models attribute smaller weights to outlying observations, which produce robust parameter estimates. Also, some inferential results are presented. In addition, based on local influence and deviance component and martingale-type residuals, a diagnostics analysis is derived. Finally, a motivating example from the medical field is analyzed using log-BS regression models. Since the parameter estimates appear to be very sensitive to outlying and influential observations, the Student-t log-BS regression model should attenuate such influences. The model checking methodologies developed in this paper are used to compare the fitted models.     

Variable selection approach for zero-inflated count data via adaptive lasso

This article proposes a variable selection approach for zero-inflated count data analysis based on the adaptive lasso technique. Two models including the zero-inflated Poisson and the zero-inflated negative binomial are investigated. An efficient algorithm is used to minimize the penalized log-likelihood function in an approximate manner. Both the generalized cross-validation and Bayesian information criterion procedures are employed to determine the optimal tuning parameter, and a consistent sandwich formula of standard errors for nonzero estimates is given based on local quadratic approximation. We evaluate the performance of the proposed adaptive lasso approach through extensive simulation studies, and apply it to analyze real-life data about doctor visits.     

Confidence regions for the stationary point of a quadratic response surface based on the asymptotic distribution of its MLE

Response surface methodology aims at finding the combination of factor levels which optimizes a response variable. A second order polynomial model is typically employed to make inference on the stationary point of the true response function. A suitable reparametrization of the polynomial model, where the coordinates of the stationary point appear as the parameter of interest, is used to derive unconstrained confidence regions for the stationary point. These regions are based on the asymptotic normal approximation to the sampling distribution of the maximum likelihood estimator of the stationary point. A simulation study is performed to evaluate the coverage probabilities of the proposed confidence regions. Some comparisons with the standard confidence regions due to Box and Hunter are also showed.     

A Simple Approximate Procedure for Constructing Binomial and Poisson Tolerance Intervals

The problems of constructing tolerance intervals for the binomial and Poisson distributions are considered. Closed-form approximate equal-tailed tolerance intervals (that control percentages in both tails) are proposed for both distributions. Exact coverage probabilities and expected widths are evaluated for the proposed equal-tailed tolerance intervals and the existing intervals. Furthermore, an adjustment to the nominal confidence level is suggested so that an equal-tailed tolerance interval can be used as a tolerance interval which includes a specified proportion of the population, but does not necessarily control percentages in both tails. Comparison of such coverage-adjusted tolerance intervals with respect to coverage probabilities and expected widths indicates that the closed-form approximate tolerance intervals are comparable with others, and less conservative, with minimum coverage probabilities close to the nominal level in most cases. The approximate tolerance intervals are simple and easy to compute using a calculator, and they can be recommended for practical applications. The methods are illustrated using two practical examples.