The dynamic planning of a condition-based maintenance strategy based on the degradation process has been of great interest in the last decade. This study considers a general and flexible lifetime delayed heterogeneous degradation process (LDHDP), in which the degradation process has random initial effects and covariates associated with the operation characteristic. This model is suitable for engineering data that indicate a product's degradation process starts randomly, and the initiation time of the degradation process is correlated with the degradation rate (i.e., the heterogeneity of the degradation process). Based on the LDHDP, we develop a periodic-sequential inspection and maintenance strategy that minimizes the expected unit time cost. Considering the heterogeneity between products, the value of information is employed to evaluate the information value of the subsequent inspection time, which is used to adapt different degradation rates among products. The performance of the proposed strategy is evaluated and compared to some existing maintenance strategies using a Monte Carlo simulation study. The simulation results show the proposed strategy gives a lower cost in many scenarios. A practical example illustrates the proposed strategy, which prefers preventive maintenance more often.

Accelerated destructive degradation testing (ADDT) has become an invaluable method in reliability analysis, especially for highly reliable products. A common characteristic in many degradation studies is the presence of randomness in the initial degradation levels of testing units. Products with poor initial degradation levels tend to fail earlier. This study proposes an extended gamma process model that accommodates the random initial degradation value to accurately describe the degradation process over time. Under this modeling approach, we propose approximation methods for the conditional mean-time-to-failure (MTTF) and conditional variance of failure times to evaluate the impacts of initial degradation levels on product quality and reliability. We adopt a maximum likelihood approach to estimate the model parameters and MTTF under normal use conditions. In addition, we determine the optimal initial degradation threshold for removing poor-quality products and the proportion of products below this threshold. Based on the proposed model, the optimal ADDT plan is derived by minimizing the asymptotic variance of estimated MTTF under normal use conditions. A Monte Carlo simulation is conducted to assess the performance of the proposed inferential methods. Finally, a real-world ADDT dataset is analyzed to illustrate the proposed model and methodologies for making informed decisions on quality and reliability management.

Multiple performance characteristics (PCs) are common in modern products with complex structures and diverse functions. These PCs are usually dependent, with significant unit-specific variability among the multivariate degradation processes. Therefore, the associated degradation modeling for dependent multivariate degradation processes is important. This article proposes a novel multivariate$t$degradation model for this purpose. Specifically, the dependence between multivariate degradation processes is captured by random drift parameters that follow a multivariate normal distribution, and the variation in diffusion parameters and variance–covariance is characterized by a gamma distribution. An expectation-maximization (EM) algorithm is employed for likelihood inference, and confidence intervals of the model parameters are constructed by normal approximation and bootstrap method. A theoretical exploration investigating the effects of model misspecification in multivariate degradation modeling is addressed. Monte Carlo simulation studies are performed to validate the effectiveness of the EM algorithm and the theoretical properties of the multivariate$t$model. Finally, two illustrative examples are used to demonstrate the applicability and advantages of the proposed methods.

Data-driven methods for predicting remaining useful life (RUL) have received considerable attention in the field of degradation data analysis. The transfer learning (TL) method offers new possibilities for RUL tasks in various operational settings. However, in many engineering applications, challenges in TL arise mainly from the scarcity or high cost of labeled data in the target domain, coupled with incomplete degradation of RUL samples within the target domain. This article proposes an innovative model named deep cross-domain transfer learning for interpretable prediction The model effectively harnesses the advantages of domain adaptation (DA) techniques in mitigating domain distribution disparities and also uses the exceptional visualization capabilities inherent in the variational autoencoder (VAE) model. This method integrates the VAE framework with regression networks and utilizes DA techniques to align feature spaces, achieving cross-domain RUL prediction with unlabeled target domain data and cross-domain visualization of the entire degradation process. The reproducing kernel Hilbert space is considered in domain adaption to control the complexity of hypothesis space. The effectiveness of the proposed method is demonstrated by analyzing the real C-MAPSS dataset.

Predicting remaining useful life (RUL) plays a crucial role in predictive maintenance, improving system reliability, availability, and safety. However, obtaining data from the target domain is often challenging in real-world industrial applications. This article focuses on the domain generalization (DG) problem, where the attention is directed toward adapting algorithms to unseen domains. Building upon the popular algorithm domain adversarial neural network (DANN) for DG, we extend the contrastive adversarial domain adaptation method using a multiple source–source adversarial network to learn domain-invariant features from multiple source domains. In addition, we incorporate the swin-transformer structure into our model to enhance its capability in extracting time–frequency features, leveraging its excellent performance in visual DG problems. Furthermore, to expand the training dataset, we propose a novel augmentation algorithm for time–frequency data. Through predictive experiments in scenarios with unknown domain labels, we validate the contribution of the proposed methods to RUL prediction performance.

The evaluation of long-lifetime and high-reliability products has attracted much attention. Stochastic degradation modeling is one of the most popular methods. The classical stochastic processes are frequently employed to discuss degradation trajectories. Most current work assumes that the underlying probability model of a degradation process is known or fixed in the estimation and model selection procedures. However, the ground-truth degradation model is usually unavailable in engineering applications. This article proposes a feasible parameter estimation and model selection procedure by measuring the distribution divergence among the nonparametric estimated model and some candidate models. In the proposed methods, it is not necessary to assume the availability of a ground-true model, which is replaced by a nonparametric estimated model. The proposed methodologies are suitable for restricted independent and nonidentically distributed samples. We discuss the large sample property of the suggested estimators. We report the Monte Carlo simulation study and practical data analysis to demonstrate our methods.

Global market competition has led to manufacturers encountering a decision problem as the reliability of their products exceeds a given standard. Acceptance sampling tests (ASTs) can be expensive and time-consuming, especially for products with high reliability or quality. Thus, careful planning of the acceptance rule and determination of a suitable sample size for the test are important. In this article, an exact method for the optimal planning of ASTs is proposed for cases in which the product lifetimes follow a distribution in the general log-location-scale family of distributions. A novel procedure using the pivotal quantity is established based on the method-of-moments estimators of the model parameters, and the distribution of the reliability estimator is derived. An algorithm for obtaining the optimal sample size and the corresponding acceptability constant is presented in which the producer's and consumer's risks are constrained according to certain acceptance criteria. We apply the proposed method to a censored AST by generating censored observations with a quantile-filling method under Type-II censoring. The performance of the proposed AST with a complete or censored sample is studied and compared with an AST based on the maximum likelihood method. The robustness of the proposed AST is also examined under model misspecification. The results show that the proposed AST performs well under different scenarios.

Minimizing the divergence between two probability distributions offers an alternative parameter estimation method. The current literature mainly focuses on minimizing the Kullback-Leibler (K-L) divergence between the true and the proposed models in which the true model is assumed to be known or fixed. In this paper, we propose a parameter estimation method that minimizes the$f$-divergence between two probability distributions. The method is suitable for different situations, no matter the true distribution is known or not. The statistical properties of the estimator, including consistency and asymptotic normality, are established. As an illustration, our method is employed to estimate the degradation model, which is a model frequently used to assess the lifetime of highly reliable products. A simulation study and a real degradation data analysis are presented to illustrate the effectiveness of the proposed estimation method.

Motivated by an accelerated degradation test (ADT) on the power gain of microwave power amplifiers, in this article we propose a model-ranking approach for the estimation of some important reliability characteristics. Different degradation models and statistical lifetime distributions are applied to model the data obtained from the ADT. We study the effect of model misspecification in estimating the reliability characteristics when the behavior of the degradation process and the underlying degradation-data-generating mechanism are unknown. We then propose a model-ranking approach with weighted estimation procedures when multiple candidate models are under consideration. Through a Monte Carlo simulation study, we show that the proposed approach is robust and insensitive to model misspecification. Finally, the ADT data from the motivating example are used to illustrate the proposed methodologies.

In this paper, the methods of information geometry are employed to investigate a generalized Bayes rule for prediction. Taking α-divergences as the loss functions, optimality, and asymptotic properties of the generalized Bayesian predictive densities are considered. We show that the Bayesian predictive densities minimize a generalized Bayes risk. We also find that the asymptotic expansions of the densities are related to the coefficients of the α-connections of a statistical manifold. In addition, we discuss the difference between two risk functions of the generalized Bayesian predictions based on different priors. Finally, using the non-informative priors (i.e., Jeffreys and reference priors), uniform prior, and conjugate prior, two examples are presented to illustrate the main results.

The failure of any component within a series system results in the failure of the system. Incorporating active redundancy into a system generally improves its reliability and availability. This type of system does not belong to the category of coherent systems, and thus those results on coherent systems are not applicable. Hence, it is of great interest to develop statistical methodologies that perform well for inference on reliability of redundancy systems. In a life-test for multicomponent systems, component lifetimes may not be observable, but one may observe a set of components that failed along with the system. The data available in this form are called autopsy data, and, in this case, lifetime information on some of the components is missing. In this paper, we consider an equal load-sharing model and develop an expectation-maximization algorithm for estimating system reliability characteristics based on such an autopsy data. The load-sharing parameter indicates whether incorporating active redundancy improves system reliability or induces rapid failure of the system. The performance of the proposed methodology is then evaluated through Monte Carlo simulations and then illustrated with two numerical examples.

In this paper, a mixture representation is derived for the pooled system lifetimes arising from a life-test on two or more independent samples. The components of each system are assumed to have the same common absolutely continuous distribution, but the system signature may vary between the samples. These mixtures are then used for developing exact nonparametric inference in the form of confidence intervals for quantiles of component or system lifetimes, as well as prediction intervals for future component or system lifetimes. Examples are finally provided to illustrate the developed methods. It is noted that testing with systems rather than components directly can reduce the expected number of failures while maintaining nominal coverage probability.

Statistical inference of the component lifetime distribution is developed when Type-II censored system lifetime data are observed with a known system structure. The component lifetime distributions are assumed to be from either the log-location-scale family of distributions or the location-scale family of distributions. Two estimation methods, the maximum likelihood method, and the regression-based method, are proposed for the model parameters, and the corresponding computational formulae are provided. Construction of confidence intervals for the model parameters is also considered. The methodologies are illustrated with two commonly used lifetime distributions: the Weibull, and the lognormal. Monte Carlo simulations are used to study the performances of the point and interval estimation methods proposed here. Finally, some recommendations are made based on the obtained simulation results.

In this paper, we discuss linear inference for the lifetime distribution of components based on a Type-II censored lifetime data of reliability systems with known signatures. We derive the best linear unbiased estimators (BLUE) for the parameter(s) in general scale and location-scale parameter families. The exact computational formulas of the BLUE and their variances and covariance are provided. Selected tables of the coefficients of BLUE are presented for the exponential and extreme value distributions. Using these, best linear unbiased predictors of future system failure times are discussed. Finally, two examples are provided to illustrate all the methods of inference developed here.

In this paper, the estimation of parameters based on a progressively Type-II censored sample from a modified Weibull distribution is studied. The likelihood equations, and the maximum likelihood estimators are derived. The estimators based on a least-squares fit of a multiple linear regression on a Weibull probability paper plot are compared with the MLE via Monte Carlo simulations. The observed Fisher information matrix, as well as the asymptotic variance-covariance matrix of the MLE are derived. Approximate confidence intervals for the parameters are constructed based on the s-normal approximation to the asymptotic distribution of MLE, and log-transformed MLE. The coverage probabilities of the individual s-normal-approximation confidence intervals for the parameters are examined numerically. Some recommendations are made from the results of a Monte Carlo simulation study, and a numerical example is presented to illustrate all of the methods of inference developed here.

There has been extensive research on goodness-of-fit procedures for testing whether or not a sample comes from a specified distribution. These goodness-of-fit tests range from graphical techniques, to tests which exploit characterization results for the specified underlying model. In this article, we propose a goodness-of-fit test for the location-scale family based on progressively Type-II censored data. The test statistic is based on sample spacings, and generalizes a test procedure proposed by Tiku . The null distribution of the test statistic is shown to be approximated closely by a s-normal distribution. However, in certain situations it would be better to use simulated critical values instead of the s-normal approximation. We examine the performance of this test for the s-normal and extreme-value (Gumbel) models against different alternatives through Monte Carlo simulations. We also discuss two methods of power approximation based on s-normality, and compare the results with those obtained by simulation. Results of the simulation study for a wide range of sample sizes, censoring schemes, and different alternatives reveal that the proposed test has good power properties in detecting departures from the s-normal and Gumbel distributions. Finally, we illustrate the method proposed here using real data from a life-testing experiment. It is important to mention here that this test can be extended to multi-sample situations in a manner similar to that of Balakrishnan et al.

The likelihood equations based on a progressively Type-II censored sample from a Gaussian distribution do not provide explicit solutions in any situation except the complete sample case. This paper examines numerically the bias and mean square error of the MLE, and demonstrates that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic s-normality are unsatisfactory, and particularly so when the effective sample size is small. Therefore, this paper suggests using unconditional simulated percentage points of these pivotal quantities for constructing s-confidence intervals. An approximation of the Gaussian hazard function is used to develop approximate estimators which are explicit and are almost as efficient as the MLE in terms of bias and mean square error; however, the probability coverages of the corresponding pivotal quantities based on asymptotic s-normality are also unsatisfactory. A wide range of sample sizes and progressive censoring schemes are used in this study.