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A warranty product often has an interval structure, resulting in different warranty periods for each component due to their distinct degradation paths. From the manufacturer's perspective, it is reasonable to establish appropriate warranty periods for each part to achieve cost savings. This paper introduces a dependent series configuration consisting of a main component and an auxiliary component. The failure of the main component leads to system failure, while the failure of the auxiliary component only causes a random damage to the main component. These two components follow different Wiener degradation paths, and then their first passage times (FPTs) follow disparate inverse Gaussian distributions. Assuming a pre-deterministic warranty period, W , for the main component, and η W , 0 < η < 1 , for the auxiliary component, the warranty period can be divided into two stages: [ 0 , η W ] and [ η W , W ]. In this context, the paper proposes warranty claim policies for replacement, objective-oriented imperfect repair (OOIR), and refund. The corresponding warranty costs sustained by manufacturer and consumer within these two stages are considered. Finally, a numerical example is presented to illustrate the warranty costs sustained by manufacturer and consumer under different scenarios, as well as the total warranty costs. • The degradation paths, warranty periods and warranty costs of the series system is analyzed. • The warranty costs sustained by manufacturer and consumer under different scenarios is analyzed. • The interdependence between various components is considered. [ABSTRACT FROM AUTHOR]

We obtain the strong law of large numbers, Glivenko-Cantelli theorem, central limit theorem, functional central limit theorem for various Bayesian nonparametric priors which include the stick-breaking process with general stickbreaking weights, the two-parameter Poisson-Dirichlet process, the normalized inverse Gaussian process, the normalized generalized gamma process, and the generalized Dirichlet process. For the stick-breaking process with general stick-breaking weights, we introduce two general conditions such that the central limit theorem and functional central limit theorem hold. Except in the case of the generalized Dirichlet process, since the finite dimensional distributions of these processes are either hard to obtain or are complicated to use even they are available, we use the method of moments to obtain the convergence results. For the generalized Dirichlet process we use its marginal distributions to obtain the asymptotics although the computations are highly technical. [ABSTRACT FROM AUTHOR]

Multiple sensors are commonly used for accelerated degradation monitoring. Since different sensors may be sensitive at different stages of the accelerated degradation process and each sensor dataset may contain only partial information of the unit degradation, then integration approaches of the accelerated degradation data from multiple sensors can effectively improve degradation modeling and life prediction accuracy. We present a non‐parametric approach that assigns weights to each sensor based on the dynamic clustering of the sensors' observations. Missing data are common in degradation data acquisition, especially when multiple sensors are used. We provide two approaches for data interpolation: the nonlinear Brownian bridge and the inverse Gaussian bridge when the underlying degradation paths follow the Brownian motion process and inverse Gaussian process, respectively. The stochastic bridges capture the nonlinearity and uncertainties of the degradation processes. The data integration model and stochastic bridge models are validated with real accelerated fatigue crack growth data monitored with multiple NDT sensors. The proposed models provide an accurate accelerated degradation path and reliability prediction. [ABSTRACT FROM AUTHOR]

The use of degradation data to estimate the remaining useful life (RUL) has gained great attention with the widespread use of prognostics and health management on safety critical systems. Accurate RUL estimation can prevent system failure and reduce the running risks since the efficient maintenance service could be scheduled in advance. In this paper, we present a degradation modeling and RUL estimation approach by using available degradation data for a deteriorating system. An inverse Gaussian process with the random effect is firstly used to characterize the degradation process of the system. Expectation maximization algorithm is then adopted to estimate the model parameters, and the random parameters in the degradation model are updated by Bayesian method, which makes the estimated RUL able to be real-time updated in terms of the fresh degradation data. Our proposed method can capture the latest condition of the system by means of updating degradation data continuously, and obtain the explicit expression of RUL distribution. Finally, a numerical example and a practical case study are provided to show that the presented approach can effectively model degradation process for the individual system and obtain better results for RUL estimation. [ABSTRACT FROM AUTHOR]

Reliability estimation for key components of a mechanical system is of great importance in prognosis and health management in aviation industry. Both degradation data and failure time data contain abundant reliability information from different sources. Considering multiple variable-dependent degradation performance indicators for mechanical components is also an effective approach to improve the accuracy of reliability estimation. This study develops a bivariate-dependent reliability estimation model based on inverse Gaussian process and copulas fusing degradation data and failure time data within one computation framework. The inverse Gaussian process model is used to describe the degradation process of each performance indicator. Copula functions are used to capture the dependent relationship between the two performance indicators. In order to improve the reliability estimation accuracy, both degradation data and failure time data are used simultaneously to estimate the unknown parameters in the degradation model based on the likelihood function transformed using the zeros-ones trick. A simulation study and a real application in the reliability estimation of mechanical seal used in airborne hydraulic pump are conducted to validate the effectiveness and accuracy of the proposed model compared with existing reliability models. [ABSTRACT FROM AUTHOR]

Risk management is a popular and important problem in academia and industry. From a small-scale system, such as city logistics, to a large-scale system, such as the supply chain of a global industrial or financial system, efficient risk management is required to prevent loss from uncertainty. In this paper, we assume that risk factors follow the Lévy process, and propose a stylized model, based on regression, that can estimate the risk of a complicated system under the framework of nest simulation. Specifically, portfolio risk estimation using the Lévy process is discussed as an example. The stylized model simplifies the risk factors artificially, and provides useful basis functions to fit the portfolio loss with little computational effort. Numerical experiments showed the good performance of the stylized model in estimating risk for the Variance Gamma process and the Normal Inverse Gaussian process, which are two examples of the Lévy process. [ABSTRACT FROM AUTHOR]

The wear of a cutting tool can lead to tool failure and product quality deterioration, and thus timely maintenance of tools is crucial. Meanwhile, the wear of tools from a same population usually exhibits heterogeneous patterns. Therefore, this paper proposes a dynamic condition-based maintenance (CBM) policy for heterogeneous-wearing tools with considering the product quality deterioration caused by tool wear. The tool wear is modelled by an Inverse Gaussian (IG) process, and the wear rate is assumed to be a random variable to characterise the heterogeneity among tool wear processes. The posterior distribution of reciprocal of tool wear rate is dynamically estimated using the online wear data based on a Bayesian approach. Moreover, the impact of tool wear on product quality deterioration is modelled. The IG process is discretized into a discrete time Markov chain (DTMC). Under the frame of the DTMC, a cost function, containing product quality loss, preventive maintenance (PM) cost and corrective maintenance cost, is developed to determine the optimal PM threshold. The cost function updates dynamically with the dynamic estimation of tool wear rate and thus enables the optimal PM threshold to be dynamically revised. The effectiveness of the proposed CBM policy is demonstrated through a case study. [ABSTRACT FROM AUTHOR]

Albeit not equivalent, in many applications the gamma and the inverse Gaussian processes are treated as if they were. This circumstance makes the misspecification problem of these models interesting and important, especially when data are affected by measurement errors, since noisy/perturbed data do not allow to verify whether the selected model is actually able to adequately fit the real (hidden) degradation process. Motivated by the above considerations, in this paper we conduct a large Monte Carlo study to evaluate whether and how the presence of measurement errors affects this misspecification issue. The study is performed considering as reference models a perturbed gamma process recently proposed in the literature and a new perturbed inverse Gaussian process that share the same non‐Gaussian distributed error term. As an alternative option, we also analyze the more classical case where the error term is Gaussian distributed. We consider both the situation where the true model is the perturbed gamma and the one where it is the perturbed inverse Gaussian. Model parameters are estimated from perturbed data using the maximum likelihood method. Estimates are retrieved by using a new sequential Monte Carlo EM algorithm, which use allows to hugely mitigate the severe numerical issues posed by the direct maximization of the likelihood. The risk of incurring in a misspecification is evaluated as percentage of times the Akaike information criterion leads to select the wrong model. The severity of a misspecification is evaluated in terms of its impact on maximum likelihood estimate of the mean remaining useful life. [ABSTRACT FROM AUTHOR]

In this paper, we propose the M -optimality criterion for designing constant-stress accelerated degradation tests (ADTs). The newly proposed criterion concentrates on the degradation mechanism equivalence rather than evaluation precision or prediction accuracy which is usually considered in traditional optimization criteria. Subject to the constraints of total sample number, test termination time as well as the stress region, an optimum constant-stress ADT plan is derived by determining the combination of stress levels and the number of samples allocated to each stress level, when the degradation path comes from inverse Gaussian (IG) process model with covariates and random effects. A numerical example is presented to verify the robustness of our proposed optimum plan and compare its efficiency with other test plans. Results show that, with a slightly relaxed requirement of evaluation precision and prediction accuracy, our proposed optimum plan reduces the dispersion of the estimated acceleration factor between the usage stress level and a higher accelerated stress level, which makes an important contribution to reliability demonstration and assessment tests. [ABSTRACT FROM AUTHOR]

As the existing research of condition-based Maintenance (CBM) decision-making neglects the influence of regular detection and maintenance (RDM) on the recovery of equipment performance, it is impossible to accurately describe the state degradation characteristics and life distribution law in this case, which is not helpful to formulate reasonable and effective maintenance strategies. Aimed at this problem, a maintenance strategy combining RDM and CBM is proposed in this paper, and the performance degradation modeling and maintenance optimization model under this strategy are studied deeply. Considering the discontinuous and catastrophic performance degradation characteristics of equipment under this condition, a performance degradation model is established by using the Inverse Gaussian process from the failure mechanism. On this basis, a combined maintenance decision model constrained by risk function is constructed. The optimal maintenance cycle and preventive maintenance threshold are obtained by optimizing the equipment maintenance cost under long-term operation conditions. The relationship between the cost rate and the maintenance strategy value is obtained through the example analysis of the equipment components, and it is proved that the joint maintenance strategy can not only prolong the service life and maintenance interval of equipment, but also reduce the maintenance risk and cost. [ABSTRACT FROM AUTHOR]

In this study, a multivariate ARMA–GARCH model with fractional generalized hyperbolic innovations exhibiting fat-tail, volatility clustering, and long-range dependence properties is introduced. To define the fractional generalized hyperbolic process, the non-fractional variant is derived by subordinating time-changed Brownian motion to the generalized inverse Gaussian process, and thereafter, the fractional generalized hyperbolic process is obtained using the Volterra kernel. Based on the ARMA–GARCH model with standard normal innovations, the parameters are estimated for the high-frequency returns of six U.S. stocks. Subsequently, the residuals extracted from the estimated ARMA–GARCH parameters are fitted to the fractional and non-fractional generalized hyperbolic processes. The results show that the fractional generalized hyperbolic process performs better in describing the behavior of the residual process of high-frequency returns than the non-fractional processes considered in this study. [ABSTRACT FROM AUTHOR]

The degradation data of highly reliable products are usually analyzed by stochastic process models, such as Wiener process, gamma process and inverse Gaussian process models. If such a specific degradation model is wrongly assumed, then poor analysis results of reliability assessment would be obtained. Therefore, a class of exponential-dispersion processes, named Tweedie exponential-dispersion process (TEDP), is proposed to describe the products’ degradation paths. The TEDP model which comprises the aforementioned stochastic processes as its special cases, is more flexible and applicable for degradation modeling. Considering the nonlinear characteristics of degradation paths and the unit-to-unit variability among the product units, random-effect models are established based on the nonlinear TEDP models with random drift and dispersion parameters. To improve the mathematical tractability of these models, the variational inference, expectation maximization algorithm and differential evolution algorithm are used to estimate the unknown model parameters. Furthermore, two nonlinear TEDP models with accelerated factors and random effects are developed for accelerated degradation analysis. Finally, a simulation study and three real applications are presented to show the effectiveness and superiority of the proposed models and methods. [ABSTRACT FROM AUTHOR]

Copyright of Mohyla Mathematical Journal / Могилянський математичний журнал is the property of National University of Kyiv-Mohyla Academy, Faculty of Humanities and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We investigate multivariate subordination of Lévy processes which was first introduced by Barndorff-Nielsen et al. [O.E. Barndorff-Nielsen, F.E. Benth, and A. Veraart,Modelling electricity forward markets by ambit fields, J. Adv. Appl. Probab. (2010)], in a Hilbert space valued setting which has been introduced in Pérez-Abreu and Rocha-Arteaga [V. Pérez-Abreu and A. Rocha-Arteaga,Covariance-parameter Lévy processes in the space of trace-class operators, Infin. Dimens. Anal. Quantum Probab. Related Top. 8(1) (2005), pp. 33–54]. The processes are explicitly characterized and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space. The resulting process has the property that at each time all its finite dimensional projections are multivariate normal inverse Gaussian distributed as introduced in Rydberg [T. Rydberg,The normal inverse Gaussian Lévy process: Simulation and approximation, Commun. Stat. Stochastic Models 13 (1997), pp. 887–910]. [ABSTRACT FROM PUBLISHER]

With the increasing integration of seawater hydraulic components, much attention has been paid to the tribological characteristics of PEEK bidirectional sliding against AISI630 tribopair. In the study, a bidirectional wear model based on Inverse Gaussian (IG) process is proposed based on bidirectional wear experiment. Impact wear occurs when the tribopair changes the motion direction. Through analysing the wear process and worn surface morphology of PEEK against AISI630 tribopairs, it is found that the impact wear would accelerate the formation and uniform distribution of PEEK transfer film, and the increase of reversal frequency can improve the tribological characteristics of the bidirectional tribopairs. The IG model considering the reversal impact can accurately illustrate the performance degradation path of actual impact wear. [Display omitted] • Tribological behaviour of PEEK against AISI630 stainless steel in bidirectional sliding contact. • Friction coefficient and wear rate analysis with the reversal frequency. • Worn surface morphology and PEEK transfer film distribution under bidirectional sliding. • Bidirectional wear degradation model based on Inverse Gaussian (IG) process. [ABSTRACT FROM AUTHOR]

We consider two fractional versions of a family of nonnegative integer-valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Pólya-Aeppli process, the Poisson inverse Gaussian process, and the negative binomial process. We also define and study some more general fractional versions with two fractional parameters. [ABSTRACT FROM AUTHOR]

We provide closed-form pricing formulas for a wide variety of path-independent options, in the exponential Lévy model driven by the normal inverse Gaussian process. The results are obtained in both the symmetric and asymmetric models, and take the form of simple and quickly convergent series, under some conditions involving the log-forward moneyness and the maturity of instruments. Proofs are based on a factorized representation in the Mellin space for the price of an arbitrary path-independent payoff and on tools from complex analysis. The validity of the results is assessed thanks to several comparisons with standard numerical methods (Fourier and Fast Fourier transforms, Monte-Carlo simulations) for realistic sets of parameters. Precise bounds for the convergence speed and the truncation error are also provided. [ABSTRACT FROM AUTHOR]

Burn‐in tests have been discussed extensively in the reliability literature, wherein we operate items until high degradation values are observed, which could separate the weak units from the normal ones before they get to the market. This concept is often referred to as a screening procedure, and it involves misclassification errors. Commonly, the underlying degradation process is assumed to be a Wiener or a gamma process, based on which several optimal burn‐in policies have been developed in the literature. In this article, we consider the mixture inverse Gaussian process, which possesses monotone degradation paths and some interesting properties. Under this process, we present a decision rule for classifying a unit under test as normal or weak based on burn‐in time and a set of cutoff points. Then, an economic cost model is used to find the optimal burn‐in time and the optimal cutoff points, when the estimation of model parameters is based on an analytical method or an approximate method involving copula theory. Finally, an example of a real dataset on light amplification by stimulated emission of radiations, well known in the reliability literature, is used to illustrate the model and the inferential approach proposed here. [ABSTRACT FROM AUTHOR]

In accelerated degradation testing (ADT), test data from higher than normal stress conditions are used to find stochastic models of degradation, e.g., Wiener process, Gamma process, and inverse Gaussian process models. In general, the selection of the degradation model is made with reference to one specific product and no consideration is given to model uncertainty. In this paper, we address this issue and apply the Bayesian model averaging (BMA) method to constant stress ADT. For illustration, stress relaxation ADT data are analyzed. We also make a simulation study to compare the s\-credibility intervals for single model and BMA. The results show that degradation model uncertainty has significant effects on the p\-quantile lifetime at the use conditions, especially for extreme quantiles. The BMA can well capture this uncertainty and compute compromise s\-credibility intervals with the highest coverage probability at each quantile. [ABSTRACT FROM PUBLISHER]

Random probability measures are the main tool for Bayesian nonparametric inference, with their laws acting as prior distributions. Many well-known priors used in practice admit different, though equivalent, representations. In terms of computational convenience, stick-breaking representations stand out. In this paper we focus on the normalized inverse Gaussian process and provide a completely explicit stick-breaking representation for it. This result is of interest both from a theoretical viewpoint and for statistical practice. [ABSTRACT FROM PUBLISHER]

Copyright of Maintenance & Reliability / Eksploatacja i Niezawodność is the property of Polish Scientific & Technical Society Consumables, Polish Maintenance Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Accelerated degradation test (ADT) has become an efficient approach to evaluate the reliability for highly reliable products. However, when modeling accelerated degradation data with degradation models, it is difficult to exactly figure out the changing rules of parameters with stress variables varying. At present, the changing rules are mainly assumed according to engineering experience or subjective judgements, which probably results in inaccurately extrapolating the reliability. To figure out the changing rules of parameters with stress variables varying, the acceleration factor constant principle and its application under ADT are studied in the paper. It is well known that the acceleration factor between any two different stress levels should be a constant under an effective ADT. For each degradation model, its parameters should obey special changing rules to satisfy that the acceleration factor maintains a constant throughout an ADT. Taking three extensively used stochastic process models as examples, including Wiener process model, gamma process model, and inverse Gaussian process model, the method of deducing the changing rules of the parameters according to the acceleration factor constant principle was demonstrated. A simulation test was conducted to validate the deduced changing rules of the parameters for the three stochastic process models. An illustrative example involving self-regulating heating cables was used to illustrate the application of the acceleration factor constant principle under ADT. Results indicate that the acceleration factor constant principle offers an appealing and credible approach to help model accelerated degradation data. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Forecasting the remaining useful life (RUL) is a crucial aspect of prognostics and health management (PHM), which has garnered significant attention in academic and industrial domains in recent decades. The accurate prediction of RUL relies on the creation of an appropriate degradation model for the system. In this paper, a general representation of diffusion process models with three sources of uncertainty for RUL estimation is constructed. According to time-space transformation, the analytic equations that approximate the RUL probability distribution function (PDF) are inferred. The results demonstrate that the proposed model is more general, covering several existing simplified cases. The parameters of the model are then calculated utilizing an adaptive technique based on the Kalman filter and expectation maximization with Rauch-Tung-Striebel (KF-EM-RTS). KF-EM-RTS can adaptively estimate and update unknown parameters, overcoming the limits of strong Markovian nature of diffusion model. Linear and nonlinear degradation datasets from real working environments are used to validate the proposed model. The experiments indicate that the proposed model can achieve accurate RUL estimation results. [ABSTRACT FROM AUTHOR]

Accurately predicting the remaining useful life (RUL) of critical mechanical components is a central challenge in reliability engineering. Stochastic processes, which are capable of modeling uncertainties, are widely used in RUL prediction. However, conventional stochastic process models face two major limitations: (1) the reliance on strict assumptions during model formulation, restricting their applicability to a narrow range of degradation processes, and (2) the inability to account for potential variations in the degradation mechanism during modeling and prediction. To address these issues, we propose a novel mechanism-equivalence-based Tweedie exponential dispersion process (ME-based TEDP) for adaptive degradation modeling and RUL prediction of mechanical components. The proposed model enhances the original Tweedie exponential dispersion process (TEDP) by incorporating degradation mechanism equivalence, effectively capturing the correlation between model parameters. Furthermore, it improves prediction accuracy and interpretability by employing a dynamic testing–modeling–predicting strategy. Application of the ME-based TEDP model to high-speed rail bogie systems demonstrates its effectiveness and superiority over existing approaches. This study advances the theory of degradation modeling and significantly improves the precision of RUL predictions. [ABSTRACT FROM AUTHOR]

Photovoltaic (PV) modules, as essential components of solar power generation systems, significantly influence unit power generation costs. The service life of these modules directly affects these costs. Over time, the performance of PV modules gradually declines due to internal degradation and external environmental factors. This cumulative degradation impacts the overall reliability of photovoltaic power generation. This study addresses the complex degradation process of PV modules by developing a two-stage Wiener process model. This approach accounts for the distinct phases of degradation resulting from module aging and environmental influences. A power degradation model based on the two-stage Wiener process is constructed to describe individual differences in module degradation processes. To estimate the model parameters, a combination of the Expectation-Maximization (EM) algorithm and the Bayesian method is employed. Furthermore, the Schwarz Information Criterion (SIC) is utilized to identify critical change points in PV module degradation trajectories. To validate the universality and effectiveness of the proposed method, a comparative analysis is conducted against other established life prediction techniques for PV modules. [ABSTRACT FROM AUTHOR]

Taking the titanium alloy wing–body connection joint at the rear beam of a certain type of aircraft as the research object, this study analyzed the failure mechanism and verified the structural safety of the wing–body connection joint under actual flight loads. Firstly, this study verified the validity of the loading system and the measuring system in the test system through the pre-test, and the repeatability of the test was analyzed for error to ensure the accuracy of the experimental data. Then, the test piece was subjected to 400,000 random load tests of flight takeoffs and landings, 100,000 Class A load tests, and ground–air–ground load tests, and the test piece fractured under the ground–air–ground load tests. Lastly, the mechanism analysis and structural safety verification of the fatigue fracture of the joints were carried out by using a stereo microscope and scanning electron microscope. The results show that fretting fatigue is the main driving force for crack initiation, and the crack shows significant fatigue damage characteristics in the stable growth stage and follows Paris' law. Entering the final fracture region, the joint mainly experienced ductile fracture, with typical plastic deformation features such as dimples and tear ridges before fracture. The fatigue crack growth behavior of the joint was quantitatively analyzed using Paris' law, and the calculated crack growth period life was 207,374 loadings. This result proves that the crack initiation life accounts for 95.19% of the full life cycle, which is much higher than the design requirement of 400,000 landings and takeoffs, indicating that the structural design of this test piece is on the conservative side and meets the requirements of aircraft operational safety. This research is of great significance in improving the safety and reliability of aircraft structures. [ABSTRACT FROM AUTHOR]

The relative accessibility and simplicity of vestibular sensing and vestibular-driven control of head and eye movements has made the vestibular system an attractive subject to experimenters and theoreticians interested in developing realistic quantitative models of how brains gather and interpret sense data and use it to guide behavior. Head stabilization and eye counter-rotation driven by vestibular sensory input in response to rotational perturbations represent natural, ecologically important behaviors that can be reproduced in the laboratory and analyzed using relatively simple mathematical models. Models drawn from dynamical systems and control theory have previously been used to analyze the behavior of vestibular sensory neurons. In the Bayesian framework, which is becoming widely used in cognitive science, vestibular sense data must be modeled as random samples drawn from probability distributions whose parameters are kinematic state variables of the head. We show that Exwald distributions are accurate models of spontaneous interspike interval distributions in spike trains recoded from chinchilla semicircular canal afferent neurons. Each interval in an Exwald distribution is the sum of an interval drawn from an Exponential distribution and a Wald or Inverse Gaussian distribution. We show that this abstract model can be realized using simple physical mechanisms and re-parameterized in terms of the relevant kinematic state variables of the head. This model predicts and explains statistical and dynamical properties of semicircular canal afferent neurons in a novel way. It provides an empirical foundation for realistic Bayesian models of neural computation in the brain that underlie the perception of head motion and the control of head and eye movements. [ABSTRACT FROM AUTHOR]