Multidimensional degradation processes commonly arise in complex engineering systems such as aerospace equipment, military devices, and new-energy vehicles. These systems exhibit degradation behaviors characterized by multidimensionality, dependence, and heterogeneity. To effectively capture these features, this paper proposes a parameter-dependent multivariate Wiener process degradation model that simultaneously accounts for correlations among multiple degradation characteristics, dependencies between degradation rate and volatility, and individual heterogeneity within subsystems. A hierarchical Bayesian inference framework based on Gibbs sampling is developed for joint parameter estimation and uncertainty quantification. To address estimation bias issues in high-dimensional covariance structures, a hierarchical inverse Wishart prior is introduced, improving the accuracy of covariance estimation, particularly under small-variance conditions. Furthermore, a Monte Carlo–based reliability estimation scheme is proposed to approximate the first-passage time distribution of multidimensional systems. Extensive simulation studies demonstrate the superior estimation accuracy and credible interval coverage of the proposed method compared to traditional priors. Finally, applications to real-world engineering degradation data validate the model's practical effectiveness, showing that neglecting the dependence between degradation rate and volatility can lead to substantial bias in reliability assessments and maintenance decisions.

In industrial settings, the health state of a product is often difficult to observe directly. Instead, it is typically inferred from noisy degradation data that are related to the system’s operational condition. However, existing methods commonly neglect parameter uncertainty and lack the ability to perform real-time state estimation. To address these challenges, this article proposes a Bayesian inference framework for accurate online identification of system degradation states. Specifically, a Wiener process model with measurement noise is developed, and prior distributions are introduced to capture parameter uncertainty. In the offline training stage, historical measurement data are utilized to approximate the joint posterior distribution of the latent degradation states and model parameters via variational Bayesian methods. In the online stage, a state-space formulation is adopted to dynamically update the posterior distribution using real-time observations, enabling dynamic estimation of the degradation state. The proposed approach significantly reduces both storage and computational costs. Numerical simulations and real-world case studies demonstrate that the proposed method achieves superior performance in terms of both accuracy and efficiency.

Permanent magnet brakes (PMBs) have diverse applications in automotive, robotics, medical devices, and aerospace industries. However, evaluating the reliability of PMBs is challenging due to their complex construction and limited data availability. This article proposes a bivariate Wiener model to capture the degradation patterns of two key performance characteristics of PMBs, and introduces an objective Bayesian method to analyze degradation data with small sample sizes. The derivation of Jeffery prior and reference priors for different ordering groups, along with the verification of posterior propriety, are presented. A rejection sampling embedded Monte Carlo algorithm is employed to obtain Bayesian estimates of model parameters, reliability, and mean-time-to-failure of PMBs. Simulation study results demonstrate the superiority of the objective Bayesian method over the maximum likelihood approach, particularly when dealing with small sample sizes. Finally, the proposed method is applied to assess the reliability of PMBs.

Degradation is an important phenomenon for industrial products, which manifests as the gradually deterioration of some performance characteristics. The degradation process is often relevant to both time and usage, and indexing the degradation process merely by the time or usage cannot characterize the process accurately. Considering a stochastic usage process, this study proposes a degradation process model indexed by two scales, i.e., the time and the usage, where the degradation along the two scales are modeled as correlated nonlinear Wiener processes. We develop two simulation-based algorithms for reliability evaluation and study the model inference problems for the proposed model. The estimation procedure and the reliability assessment algorithms are validated by simulations. The performance of the proposed model is justified with an application to a real degradation dataset of outdoor coating materials, which shows that indexing the degradation process by two scales can considerably improve the degradation modeling performance.

By using in situ health information, many existing studies for online remaining useful life (RUL) prediction adopt a stochastic process-based degradation model and a computation-intensive parameter estimation method for RUL prediction of a single operating asset. Nevertheless, it is common that there are multiple assets under operation, and it would be more statistically efficient to jointly update their RULs by allowing information sharing among them for model parameter estimation. To this end, we propose a collaborative RUL prediction framework with closed-form online update. The framework is a hybrid algorithm that combines the conjugate prior for part of the model parameters and a stochastic approximation to the rest parameters. With this framework, a recursive online Bayesian algorithm is developed to jointly update the model parameters and RUL prediction using data from multiple operating assets. The effectiveness of the proposed method is demonstrated through a simulation study and two real cases.

In the field of reliability engineering, the gamma process plays an important role in modeling degradation processes. However, extracting lifetime information from product degradation observations has long been suffering from both ineffective modeling techniques and inefficient statistical inference methods. To overcome these challenges, we propose a reparameterized gamma process with random effects in this article. Compared with the classical gamma process, the proposed model has a more intuitive physical interpretation. In addition, statistical inference for the model can be readily done through the variational Bayesian algorithm. Combining with the Gauss–Hermite quadrature and the Laplace approximation, the algorithm yields closed-form variational posteriors for the proposed model. Its superiority over two other inference methods (expectation maximization and Monte Carlo Markov Chain) in terms of computational efficiency and estimation accuracy is demonstrated by simulation.

The working conditions of multicomponent systems are usually dynamic and stochastic. Reliability evaluation of such systems is challenging, since the components are generally positively correlated. Based on the cumulative exposure principle, we model the effects of the dynamic environments on the component lifetimes by a common stochastic time scale, and exponential dispersion process is utilized to describe the stochastic time scale. Then, the component lifetimes are shown to be positively quadrant dependent, and the joint survival function of the component lifetimes is derived, which includes the results of [1] as special cases. In this article, we show that neglecting either the effects of dynamic environments or the correlation among component lifetimes would underestimate the reliability of series systems and overestimate the reliability of parallel systems. We also investigate the problem of parameter redundancy of the model, and give some suggestions for data analysis. Simulation studies show that the unified model is flexible and useful for suggesting an optimal model given observed data.

The methods for analyzing degradation data are usually based on a specific degradation model. However, this may result in a large bias to estimate the reliability of the product when the assumed model is wrong. In this paper, a new stochastic process, called exponential dispersion process, is developed to describe the degradation path of the product's physical or chemical characteristics. Exponential dispersion process includes the most used degradation models (Wiener process, gamma process, inverse Gaussian process, and compound Poisson process) as special cases. Thus, it is useful for suggesting an appropriate degradation model for a specific dataset. We investigate the exponential dispersion model with nonlinear degradation path, and use approximated maximum likelihood method and bootstrap approach to obtain the point and confidence interval estimates of the unknown model parameters. Furthermore, exponential dispersion process models with acceleration factors and random effects are also explored. The developed methodologies are then applied to real data analysis.

Degradation test is an effective method in assessing product reliability when measurements of degradation leading to failure can be observed. The accuracy of reliability inference in degradation analysis highly depends on the fitted model to the observed degradation data. Sometimes, observed degradation paths of the products exhibit multiphase pattern over the testing period. In this paper, we propose a change-point Wiener process with measurement errors (CPWPME) to fit two-phase degradation paths of organic light-emitting diodes (OLEDs). We assume the unit-specific parameters of the CPWPME model by using hierarchical Bayesian method. Based upon the proposed approach, the failure-time distribution and the remaining useful life distribution along with mean time to failure and mean residual life function are derived in closed form. A simulation study shows the utility of the proposed CPWPME model and the validity of the hierarchical Bayesian approach for the degradation data possessing two-phase degradation characteristics. In the analysis of OLED degradation data, the hierarchical Bayesian CPWPME model provides higher modeling flexibility and prediction power for future testing units than existing three degradation models.

Modern products are usually designed with high reliability and have complex structures, and degradation analysis of the complex systems with two or multiple performance characteristics is still a challenge. In this paper, we propose a new bivariate degradation model based on the Wiener process. There are three main merits of the proposed model: it can describe the common factor affecting the degradation of the two performance characteristics and unit-to-unit variation simultaneously, the reliability functions of the system and the remaining useful life of the system have analytic forms, and the model parameters and the missing values can be estimated by the Bayesian method and data augmentation. The simulation study and data analysis show that the Bayesian method and the proposed model have satisfactory performance.

In addition to conventional censoring schemes such as Type-I or Type-II censoring, progressively censoring is a useful method to reduce cost and obtain additional reliability information in accelerated life testing. Statistical inference for accelerated life testing (ALT) with Weibull distribution and progressively censoring is found to be difficult. This paper develops generalized fiducial inference techniques for the constant-stress ALT model with Weibull distribution and progressively Type-II censoring. Simulation studies reveal the good performance of the proposed inference methods. An example is given for illustration.

In this paper, a Bayesian criterion is proposed based on the expected Kullback-Leibler divergence between the posterior and the prior distributions of the parameters of interest. We call the Bayesian criterion the reference optimality criterion, which is to find an optimal plan to maximize the amount of information from the data. A large-sample approximation is utilized to simplify the formula to obtain optimal plans numerically. Because optimal plans based on reference optimality criterion do not depend on the sample size, a modified reference optimality criterion is proposed. We give numerical examples using the Weibull distribution with type I censoring to illustrate the methods, and to examine the influence of the prior distribution, censoring time, and sample size. We also compare our methods with other criteria through Monte Carlo simulation.

Bayesian analysis of the series system failure data under step-stress accelerating life testing is proposed when the cause of failure may not have been identified but has only been narrowed down to a subset of all potential risks. A general Bayesian formulation is investigated for the log-location-scale distribution family that includes most commonly used parametric lifetime distributions. Reparameterization is introduced for estimating the lifetime under the use condition stress and other parameters directly. The posterior analysis is done by Markov chain Monte Carlo sampling. The methodology is illustrated through the Weibull distributions, and a numerical example.

In this paper, we consider a series system with twos-independent components, each of which has a Pareto distributed lifetime. We assume the lifetime data from the system are masked, and the masking probability is dependent. Noninformative prior, and conjugate prior are considered; and Gibbs sampling method is used to obtain the estimators of the unknown parameters, the reliability function, and the hazard function. A numeric example demonstrates how to apply the theoretical results.