Statistical Analysis of General Degradation Path Model and Failure Time Data with Multiple Failure Modes

ABSTRACT

This article considers degradation and failure time models with multiple failure modes which used to study the problem of longevity and aging in survival analysis and reliability. Degradation process is modeled using general nonparametric, nonlinear path models. Semi-parametric models for the intensities of the traumatic failures are used supposing that these intensities depend on degradation level. Semi-parametric estimators of various reliability characteristics are proposed and asymptotic properties of the estimators are obtained. The theoretical results are illustrated using simulated data.     

Joint modeling of degradation and failure time data

This paper surveys some approaches to model the relationship between failure time data and covariate data like internal degradation and external environmental processes. These models which reflect the dependency between system state and system reliability include threshold models and hazard-based models. In particular, we consider the class of degradation–threshold–shock models (DTS models) in which failure is due to the competing causes of degradation and trauma. For this class of reliability models we express the failure time in terms of degradation and covariates. We compute the survival function of the resulting failure time and derive the likelihood function for the joint observation of failure times and degradation data at discrete times. We consider a special class of DTS models where degradation is modeled by a process with stationary independent increments and related to external covariates through a random time scale and extend this model class to repairable items by a marked point process approach. The proposed model class provides a rich conceptual framework for the study of degradation–failure issues.     

Inference from Accelerated Degradation and Failure Data Based on Gaussian Process Models

An important problem in reliability and survival analysis is that of modeling degradation together with any observed failures in a life test. Here, based on a continuous cumulative damage approach with a Gaussian process describing degradation, a general accelerated test model is presented in which failure times and degradation measures can be combined for inference about system lifetime. Some specific models when the drift of the Gaussian process depends on the acceleration variable are discussed in detail. Illustrative examples using simulated data as well as degradation data observed in carbon-film resistors are presented.     

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.     

A combinatorial estimation approach for storage reliability with initial failures based on periodic testing data

Storage reliability that measures the ability of products in a dormant state to keep their required functions is studied in this paper. Unlike the operational reliability, storage reliability for certain types of products may not be always 100% at the beginning of storage since there are existing possible initial failures that are normally neglected in the models of storage reliability. In this paper, a new combinatorial approach, the nonparametric measure for the estimates of the number of failed products and the current reliability at each testing time in storage, and the parametric measure for the estimates of the initial reliability and the failure rate based on the exponential reliability function, is proposed for estimating and predicting the storage reliability with possible initial failures. The proposed method has taken into consideration that the initial failure and the reliability testing data, before and during the storage process, are available for providing more accurate estimates of both initial failure probability and the probability of storage failures. When storage reliability prediction that is the main concern in this field should be made, the nonparametric estimates of failure numbers can be used into the parametric models for the failure process in storage. In the case of exponential models, the assessment and prediction method for storage reliability is provided in this paper. Finally, numerical examples are given to illustrate the method. Furthermore, a detailed comparison between the proposed method and the traditional method, for examining the rationality of assessment and prediction on the storage reliability, is presented. The results should be useful for planning a storage environment, decision-making concerning the maximum length of storage, and identifying the production quality.     

Reliability modeling of systems with two dependent degrading components based on gamma processes

Abstract

Many engineering systems have multiple components with more than one degradation measure which is dependent on each other due to their complex failure mechanisms, which results in some insurmountable difficulties for reliability work in engineering. To overcome these difficulties, the system reliability prediction approaches based on performance degradation theory develop rapidly in recent years, and show their superiority over the traditional approaches in many applications. This paper proposes reliability models of systems with two dependent degrading components. It is assumed that the degradation paths of the components are governed by gamma processes. For a parallel system, its failure probability function can be approximated by the bivariate Birnbaum–Saunders distribution. According to the relationship of parallel and series systems, it is easy to find that the failure probability function of a series system can be expressed by the bivariate Birnbaum–Saunders distribution and its marginal distributions. The model in such a situation is very complicated and analytically intractable, and becomes cumbersome from a computational viewpoint. For this reason, the Bayesian Markov chain Monte Carlo method is developed for this problem that allows the maximum likelihood estimates of the parameters to be determined in an efficient manner. After that, the confidence intervals of the failure probability of systems are given. For an illustration of the proposed model, a numerical example about railway track is presented.     

Parametric Multiplicative Intensities Models Fitted to Bus Motor Failure Data

Classes of two- and three-parameter multiplicative intensities models are presented. These have the advantage that they can be simply fitted by standard software. The two-parameter models include the Weibull and extreme value models, whereas the three-parameter models can have bathtub-shaped forms. The bus motor failure data are reanalysed to show that all five failures of the 191 buses can be adequately described by the same two-parameter multiplicative intensity model based on the squared logarithm of distance travelled in service since previous repair. This is a new member of the multiplicative intensity family, based on the squared logarithm of distance, or of time for survival data.     

Statistical planning and inference in accelerated life testing using the CHSS model

Failures of highly reliable units are rare and it may be not possible to gather the failure time data needed for reliability estimation. One way of obtaining failures during the time given for experiments is to apply methods of accelerated life testing (ALT). In ALT units are tested at higher than usual (design) stress conditions. The purpose is to give estimators of the main reliability characteristics of units functioning under the usual stress using data of accelerated experiments. To treat such data accelerated life models are used. Here we consider special plans of experiments and the statistical analysis of the ALT data by numerical methods and simulation using the changing shape and scale (CHSS) model proposed by Bagdonavičius and Nikulin (1999). The CHSS model is a natural extension of the standard accelerated failure time (AFT) model. We give parametric and semiparametric estimation procedures for the CHSS model and a goodness-of-fit test for the AFT model.     

Bayesian estimation of reliability using time-to-failure distribution of parametric degradation models

This paper focusses on computing the Bayesian reliability of components whose performance characteristics (degradation – fatigue and cracks) are observed during a specified period of time. Depending upon the nature of degradation data collected, we fit a monotone increasing or decreasing function for the data. Since the components are supposed to have different lifetimes, the rate of degradation is assumed to be a random variable. At a critical level of degradation, the time to failure distribution is obtained. The exponential and power degradation models are studied and exponential density function is assumed for the random variable representing the rate of degradation. The maximum likelihood estimator and Bayesian estimator of the parameter of exponential density function, predictive distribution, hierarchical Bayes approach and robustness of the posterior mean are presented. The Gibbs sampling algorithm is used to obtain the Bayesian estimates of the parameter. Illustrations are provided for the train wheel degradation data.     

Joint modeling of linear degradation and failure time data with masked causes of failure under simple step-stress test

In this paper, we propose a method to model the relationship between degradation and failure time for a simple step-stress test where the underlying degradation path is linear and different causes of failure are possible. It is assumed that the intensity function depends only on the degradation value. No assumptions are made about the distribution of the failure times. A simple step-stress test is used to induce failure experimentally and a tampered failure rate model is proposed to describe the effect of the changing stress on the intensities. We assume that some of the products that fail during the test have a cause of failure that is only known to belong to a certain subset of all possible failures. This case is known as masking. In the presence of masking, the maximum likelihood estimates of the model parameters are obtained through the expectation–maximization algorithm by treating the causes of failure as missing values. The effect of incomplete information on the estimation of parameters is studied through a Monte-Carlo simulation. Finally, a real-world example is analysed to illustrate the application of the proposed methods.     

A Bivariate Distribution for the Reliability Analysis of Failure Data in Presence of a Forewarning or Primer Event

This article presents a bivariate distribution for analyzing the failure data of mechanical and electrical components in presence of a forewarning or primer event whose occurrence denotes the inception of the failure mechanism that will cause the component failure after an additional random time. The characteristics of the proposed distribution are discussed and several point estimators of parameters are illustrated and compared, in case of complete sampling, via a large Monte Carlo simulation study. Confidence intervals based on asymptotic results are derived, as well as procedures are given for testing the independence between the occurrence time of the forewarning event and the additional time to failure. Numerical applications based on failure data of cable insulation specimens and of two-component parallel systems are illustrated.     

Weighted Estimator for the Linear Transformation Models with Multivariate Failure Time Data

In this article, a simple and efficient weighted method is proposed to improve the estimation efficiency for the linear transformation models with multivariate failure time data. Asymptotic properties of the estimators with a closed-form variance-covariance matrix are established. In addition, a goodness-of-fit test is developed to evaluate the adequacy of the model. The performance of proposed method and the comparison on the efficiency between the proposed method and the working independence method (Lu, 2005) are conducted in finite-sample situation by simulation studies. Finally a real data set from the Busselton Population Health Surveys is illustrated to validate the proposed methodology. The related proofs of the theorems are given in the Appendix.     

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.     

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.     

Reliability Research of Dependent Failure Systems Using Copula

The computation of reliability characteristics of a system that consists of dependent components sometimes becomes difficult, especially when a specific type of dependence is not identified. In this paper, some systems with arbitrary dependent components are studied using copula. In the system, the components are dependent on each other and the dependent relations may be either linear or nonlinear correlation. The efficient formulas are presented to compute the reliability characteristics, such as reliability function, failure rate and meantime to failure of series, parallel and k-out-of-n systems. The reliability functions of dependant systems are compared with independent system. At last, the numerical examples are presented to illustrate the results obtained in this paper.     

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.     

Properties of test statistics applied to residuals in failure time models

Asymptotic properties of a class of test statistics when applied to hazard-based residuals arising in survival and reliability models are presented. These test statistics are useful in goodness-of-fit tests and model validation. The properties are obtained by examining the asymptotic properties of generalized residual processes, which are (possibly random) time transformations of the processes associated with the incomplete failure times. Since the time transformations depend on unknown model parameters, the residual processes are obtained by replacing the unknown parameters by their estimators. The results shed light on the effects of estimating parameters to obtain the residual processes. Implications concerning possible pitfalls of some existing model validation procedures utilizing hazard-based residuals and ways to correct these problems are discussed.     

Marginal Analysis For Clustered Failure Time Data

Clustered failure time data are commonly encountered in biomedical research where the study subjects from the same cluster (e.g., family) share the common genetic and/or environmental factors such that the failure times within the same cluster are correlated. Two approaches that are commonly used to account for the intra-cluster association are frailty models and marginal models. In this paper, we study the marginal proportional hazards model, where the structure of dependence between individuals within a cluster is unspecified. An estimation procedure is developed based on a pseudo-likelihood approach, and a risk set sampling method is proposed for the formulation of the pseudo-likelihood. The asymptotic properties of the proposed estimators are studied, and the related issues regarding the statistical efficiencies are discussed. The performances of the proposed estimator are demonstrated by the simulation studies. A data example from a child vitamin A supplementation trial in Nepal (Nepal Nutrition Intervention Project-Sarlahi, or NNIPS) is used to illustrate this methodology.     

Bivariate degradation modelling with marginal heterogeneous stochastic processes

In this paper, we consider that the degradation of two performance characteristics of a product can be modelled by stochastic processes and jointly by copula functions, but different stochastic processes govern the behaviour of each performance characteristic (PC) degradation. Different heterogeneous and homogeneous models are presented considering copula functions and different combinations of the most used stochastic processes in degradation analysis as marginal distributions. This is an important aspect to consider because the behaviour of the degradation of each PC may be different in its nature. As the joint distributions of the proposed models result in complex distributions, the estimation of the parameters of interest is performed via MCMC. A simulation study is performed to compare heterogeneous and homogeneous models. In addition, the proposed models are implemented to crack propagation data of two terminals of an electronic device, and some insights are provided about the product reliability under heterogeneous models.     

Nonparametric Test for Doubly Interval-Censored Failure Time Data

This paper considers comparison of discrete failure time distributions when the survival time of interest measures elapsed time between two related events and observations on the occurrences of both events could be interval-censored. This kind of data is often referred to as doubly interval-censored failure time data. If the occurrence of the first event defining the survival time can be exactly observed, the data are usually referred to as interval-censored data. For the comparison problem based on interval-censored failure time data, Sun (1996) proposed a nonparametric test procedure. In this paper we generalize the procedure given in Sun (1996) to doubly interval-censored data case and the generalized test is evaluated by simulations.