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In industry, many highly reliable products possess multiple performance characteristics (PCs) and they typically degrade simultaneously. When such PCs are governed by a common failure mechanism or influenced by a shared operating environmental condition, interdependence between these PCs arises. To model such dependence, this article proposes a novel multivariate reparameterized inverse Gaussian (rIG) process model. It utilizes an additive structure; that is, the degradation of each marginal PC is considered as the result of the sum of two independent rIG processes, with one capturing the shared common effects across all PCs and the other describing the intrinsic randomness specific to that PC. The model has some nice statistical properties, and the system lifetime distribution can be conveniently approximated. An expectation-maximization algorithm is proposed for estimating the model parameters, and a parametric bootstrap method is designed to derive the confidence intervals. Comprehensive numerical simulations are conducted to validate the performance of the inference method. Two case studies are thoroughly investigated to demonstrate the applicability of the proposed methodology. for this article are available online. [ABSTRACT FROM AUTHOR]

Qu, Dassios, and Zhao (2021) suggested an exact simulation method for tempered stable Ornstein–Uhlenbeck processes, but their algorithms contain some errors. This short note aims to correct their algorithms and conduct some numerical experiments. [ABSTRACT FROM AUTHOR]

The step-stress accelerated degradation test (ADT) provides a feasible method for assessing the storage life of high-reliability, long-life products. However, this method results in a slower rate of performance degradation at the beginning of the test, significantly reducing the test efficiency. Therefore, this article proposes an assessment method for the step-down stress ADT that considers random effects and detection errors (SDRD). Firstly, a new Inverse Gaussian (IG) model is proposed. The model introduces the Gamma distribution to characterize the randomness of the product degradation path and uses the normal distribution to describe the detection errors of performance parameters. In addition, to solve the problem that the likelihood function of the IG model is complex and has no explicit expression, the Monte Carlo (MC) method is used to estimate unknown parameters of the model. This approach enhances computational accuracy and efficiency. Finally, to verify the effectiveness of the SDRD method, it is applied to the step-down stress ADT data from a specific missile tank to assess its storage life. Comparing the life assessment results of different methods, the conclusion shows that the SDRD method is more effective for assessing the storage life of high-reliability, long-life products. [ABSTRACT FROM AUTHOR]

The accelerated Degradation testing (ADT) experiments are important technical methods in reliability studies. Different type of accelerating degradation models has developed with the time and can be used in different types of situations. However, it has become necessary for the manager to test how many numbers of unit should be tested at a particular stress level so that the cost of testing is less. Accelerated Degradation testing (ADT) is preferred to be used in mechanized industries to obtain the required information about the reliability of product components and materials in a short period of time. Accelerated test conditions involve higher than usual pressure, temperature, voltage, vibration or any other combination of them. Data collected at such accelerated conditions are extrapolated through a physically suitable statistical model to estimate the lifetime distribution at design condition stress the life data collected from the high stresses the need to be extrapolated to estimate the life distribution under the normal-use condition. A special class of the ADT is the step-stress testing which regularly increases the stress levels at some pre-fixed time points until the test unit fails. Such experiments allow the experimenter to run the test units at higher-than-usual stress conditions in order to secure failures more quickly. The Inverse Gaussian process is flexible in incorporating random effects and explanatory variables. The different types of models based on IG process are random drift model, random volatility model and random drift-volatility model. In this paper we have considered random drift model for the study on stochastic degradation models for simple step-stress model using inverse Gaussian process observed in degradation problems. [ABSTRACT FROM AUTHOR]

A system of n$$ n $$ components is here considered, with component deterioration modeled by non decreasing time‐scaled Lévy processes. When a component fails, a sudden change in the time‐scaling functions of the surviving components is induced, which makes the components stochastically dependent. We compute the reliability function of coherent systems under this new dependence model. We next study the distribution of the ordered failure times, and establish some positive dependence properties. We also provide stochastic comparison results in the usual multivariate stochastic order between failure times of two dependence models with different parameters. Finally, some numerical experiments illustrate the theoretical results. [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]

Accelerated degradation tests (ADTs) are widely used to assess lifetime information under normal use conditions for highly reliable products. For the accelerated tests, two basic assumptions are that changing stress levels does not affect the underlying distribution family and that there is stochastic ordering for the life distributions at different stress levels. The acceleration invariance (AI) principle for ADTs is proposed to study these fundamental assumptions. Using the AI principle, a theoretical connection between the model parameters and the accelerating variables is developed for Hougaard processes. This concept can be extended to heterogeneous gamma and inverse Gaussian processes. Simulation studies are presented to support the applicability and flexibility of the Hougaard process using the AI principle for ADTs. A real data analysis using the derived relationship is used to validate the AI principle for accelerated degradation analysis. All technical details, simulation results for nonlinear cases and model mis‐specification analysis are available online as Supporting Information. [ABSTRACT FROM AUTHOR]

In response to the issue of neglecting the inherent differences in samples in the traditional degradation analysis of long-life aerospace products, a degradation modeling method combining a frailty model is proposed to quantify the unobservable heterogeneity in random degradation processes. Firstly, the frailty term is described using a generalized inverse Gaussian distribution to more comprehensively capture random effects. Secondly, an inverse Gaussian degradation model is established, combined with the frailty model to consider the unobservable heterogeneity in practical use. Through maximum likelihood estimation, parameter estimation is carried out, and Bayesian theory is used to infer an independent frailty term in order to quantify differences between individual products. Finally, reliability analysis is conducted on a core chamber cooling control valve of a certain aerospace engine. The results indicate that neglecting the unobservable heterogeneity will lead to overly ideal reliability estimates, and quantifying random effects reasonably can make the model’s estimation closer to the actual situation.

Accelerated life test provides a viable method for assessing the life of complex electromechanical products. However, it is difficult to directly assess their storage life using the failure count for complex electromechanical products with few or no failures during storage life tests. Therefore, based on the step-stress accelerated degradation test, this article proposes a new storage life assessment method (RDSA), which is particularly suitable for the storage life assessment of complex electromechanical products based on performance degradation. Firstly, this article proposes a performance degradation model that comprehensively considers the heterogeneity between units and detection errors. This model is based on the Inverse Gaussian (IG) model and introduces unit-specific random effects to capture the heterogeneity and monotonicity of product degradation paths. Additionally, it uses a normal distribution to address the challenges posed by detection errors in data analysis, thereby enhancing the accuracy of life assessment for the product. Secondly, a Quasi-Monte Carlo method (QMC) based on distribution transformation is proposed to estimate the model parameters and reliability. This method overcomes the limitations of traditional Monte Carlo methods (MC), and can significantly reduce errors, decrease computation, and accelerate convergence. Finally, the proposed method is applied to performance degradation data of the electromechanical fuzes to verify its robustness and applicability. By comparing the proposed method with other more advanced methods, the results indicate that this method is more suitable for the storage life assessment of complex electromechanical products.

The unit heterogeneity of products and the nonlinear parameter-stress relationship often exist in practice. Therefore, considering the unit heterogeneity, the nonlinear accelerated model and inverse Gaussian process are developed to depict the accelerated degradation data. On the other hand, this more realistic model leads a challenge to derive the model parameter interval estimation. Thereby, a novel two-step interval estimation method is proposed for the proposed accelerated degradation model. First, generalized confidence intervals of the parameters characterizing random effect are derived from the Cornish–Fisher expansion, and their cumulative distribution functions are obtained. Then, using the generalized pivotal quantity procedure, generalized confidence intervals of the accelerated model parameters are derived. In addition, generalized confidence intervals of predictive reliability indexes are derived to guide the practical application. Finally, simulation studies and two real examples on Spiral springs and integrated circuit devices are presented to demonstrate the implementation of the proposed method. • An Inverse Gaussian degradation model with unit heterogeneity is developed. • The nonlinear accelerated model is used to fit the stress-parameter relationship. • Generalized confidence intervals of the accelerated model parameters are derived. • A two-step interval estimation method for the proposed model is given. • Simulations and real case studies show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Degradation tests are used when there is a quality characteristic related to the life of a product. In this paper, we investigate the model mis-specification effect on the estimation precision of product's mean time to failure (MTTF) and consider a degradation test design problem. The Wiener and inverse Gaussian (IG) processes are two possible models considered. We derive expressions for the mean and variance of the estimated product's MTTF when the true model is an IG process, but is wrongly fitted by a Wiener process. We further discuss the experimental design problem and derive the explicit functional form of the estimation variances. Using the derived functions, optimal degradation test plans assuming a Wiener process model is correctly or wrongly specified are both proposed. The derived plans are applied to a laser data example. We evaluate the test efficiency of the plans derived from a Wiener process assumption when the model is mis-specified. For many optimization criteria, we observe that the obtained plans are robust even when the fitted model is mis-specified. For some criteria that may result in very different test plans under different models, we use a weighted ratio criterion to find practically useful degradation plans under model uncertainty. [ABSTRACT FROM AUTHOR]

The step-stress accelerated degradation test (ADT) provides a feasible method for assessing the storage life of high-reliability, long-life products. However, this method results in a slower rate of performance degradation at the beginning of the test, significantly reducing the test efficiency. Therefore, this article proposes an assessment method for the step-down stress ADT that considers random effects and detection errors (SDRD). Firstly, a new Inverse Gaussian (IG) model is proposed. The model introduces the Gamma distribution to characterize the randomness of the product degradation path and uses the normal distribution to describe the detection errors of performance parameters. In addition, to solve the problem that the likelihood function of the IG model is complex and has no explicit expression, the Monte Carlo (MC) method is used to estimate unknown parameters of the model. This approach enhances computational accuracy and efficiency. Finally, to verify the effectiveness of the SDRD method, it is applied to the step-down stress ADT data from a specific missile tank to assess its storage life. Comparing the life assessment results of different methods, the conclusion shows that the SDRD method is more effective for assessing the storage life of high-reliability, long-life products.

Simultaneous degradation of multiple dependent performance characteristics (PCs) is a common phenomenon for industrial products. The associated degradation modeling is of practical importance yet challenging. The dependence of the PCs can usually be attributed to two sources, one being the overall system health status and the other the common operating environments. Based on the observation, this study proposes a parsimonious multivariate Wiener process model whose number of parameters increases linearly with the dimension. We introduce a common stochastic time scale shared by all the PCs to model the dependence from the dynamic operating environment. Conditional on the time scale, the degradation of each PC is modeled as the sum of two independent Wiener processes, where one represents the common effects shared by all the PCs, and the other represents degradation caused by randomness unique to this PC. An EM algorithm is developed for model parameter estimation, and extensive simulations are implemented to validate the proposed model and the algorithms. For efficient reliability evaluation under a multivariate degradation model, including the proposed one, a bridge sampling-based algorithm is further developed. The applicability and the advantages of the proposed methods are demonstrated using a multivariate degradation dataset of a coating material. for this article are available online. [ABSTRACT FROM AUTHOR]

The scoring processes of home and away team in basketball games are modeled by two dependent inverse Gaussian processes with a team-specific parameter that measures the team strength. A common latent variable that measures the game pace is designed to characterize the dependence. A moment estimation method combined with maximum likelihood estimation is proposed to fit the parameters and a Bayesian method is applied to update the estimation and make in-game predictions. It is shown that the proposed model can obtain the same performance as the benchmark model, Gamma process model, in outcome prediction, point spread betting and model gambling. [ABSTRACT FROM AUTHOR]

The stochastic modelling of degradation processes requires different characteristics to be considered, such that it is possible to capture all the possible information about a phenomenon under study. An important characteristic is what is known as the drift in some stochastic processes; specifically, the drift allows to obtain information about the growth degradation rate of the characteristic of interest. In some phenomenon’s the growth rate cannot be considered as a constant parameter, which means that the rate may vary from trajectory to trajectory. Given this, it is important to study alternative strategies that allow to model this variation in the drift. In this paper, several hazard rate functions are integrated in the inverse Gaussian process to describe its drift in the aims of individually characterize degradation trajectories. The proposed modelling scheme is illustrated in two case studies, from which the best fitting model is selected via information criteria, a discussion of the flexibility of the proposed models is provided according to the obtained results. [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]

In this work, we have analyzed data sets from various fields using a coupled Ornstein–Uhlenbeck (OU) system of equations driven by Lévy processes. The Ornstein–Uhlenbeck model is well known for its ability to capture stochastic behaviors when used as a predictive model. There's empirical evidence showing that there exist dependencies or correlations between events; thus, we may be able to model them together. Here we show such correlation between data from finance, geophysics and health as well as show the predictive performance when they are modeled with a coupled Ornstein–Uhlenbeck system of equations. The results show that the solution to the stochastic system provides a good fit to the data sets analyzed. In addition by comparing the results obtained when the BDLP is a Γ (a , b) process or an IG(a,b) process, we are able to deduce the best choice out of the two to model our data sets. [ABSTRACT FROM AUTHOR]

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]

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.

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]