Return of products within the warranty coverage induces additional cost and loss of reputation to manufacturers. It is of practical interest to predict the return rate by experimental means before introducing a product to the market. In this paper, we propose to optimize accelerated reliability tests to achieve the goal within limited time. To describe the heterogeneity in the customers’ usage mode, a discrete random variable is employed to model the degradation rate in addition to the continuous stress variable. To further characterize the heterogeneity in the customers’ behavior, two models of product return are investigated: one assumes that customers return products once the degradation level reaches the minimum eligible return threshold and the other assumes that the threshold varies among different customers. Optimal reliability tests are planned under the large-sample assumption with two novel test schemes: global optimal planning and stress constrained planning. Insights regarding the optimal plans are gleaned to ameliorate the test planning procedure and verify the optimality. A real example from the battery industry is then presented along with the simulation study and sensitivity analysis to demonstrate the methods. We find that the randomness in return level results in different test plans. Furthermore, the constrained optimal plans offer more robustness to the compromise plans.

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Multinomial logistic regression models are popular in multicategory classification analysis, but existing models suffer several intrinsic drawbacks. In particular, the parameters cannot be determined uniquely because of the over-specification. Although additional constraints have been imposed to refine the model, such modifications can be inefficient and complicated. In this paper, we propose a novel and efficient simplex-based multinomial logistic regression technique, seamlessly connecting binomial and multinomial cases under a unified framework. Compared with existing models, our model has fewer parameters, is free of any constraints, and can be solved efficiently using the Fisher scoring algorithm. In addition, the proposed model enjoys several theoretical advantages, including Fisher consistency and sharp comparison inequality. Under mild conditions, we establish the asymptotical normality and convergence for the new model, even when the numbers of categories and covariates increase with the sample size. The proposed framework is illustrated by means of extensive simulations and real applications.

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The multicategory support vector machine (MSVM) has been widely used for multicategory classification. Despite its widespread popularity, regular MSVM cannot provide direct probabilistic results and suffers from excessive computational cost, as it is formulated on the hinge loss function and it solves a sum-to-zero constrained quadratic programming problem. In this study, we propose a general refinement of regular MSVM, termed as the simplex-based proximal MSVM (SPMSVM). Our SPMSVM uses a novel family of squared error loss functions in place of the hinge loss and it removes the explicit sum-to-zero constraint by the simplex structure. Consequently, the SPMSVM only requires solving an unconstrained linear system, leading to closed-form solutions. In addition, the SPMSVM can be cast into a weighted regression problem so that it is scalable for large-scale applications. Moreover, the SPMSVM naturally yields an estimate of the conditional category probability, which is more informative than regular MSVM. Theoretically, the SPMSVM is shown to include many existing MSVMs as its special cases, and its asymptotic and finite-sample statistical properties are well established. Simulations and real examples show that the proposed SPMSVM is a stable, scalable and competitive classifier.

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The gamma distribution is one of the most important parametric models in probability theory and statistics. Although a multitude of studies have theoretically investigated the properties of the gamma distribution in the literature, there is still a serious lack of tailored statistical tools to facilitate its practical applications. To fill the gap, this paper develops a comprehensive R package for the gamma distribution. In specific, the R package focuses on the following three important tasks: generate the gamma random variables, estimate the model parameters, and construct statistical limits, including confidence limits, prediction limits, and tolerance limits based on the gamma random variables. The proposed package encompasses the state-of-the-art methods of the gamma distribution in the literature and its usage is illustrated by a real application.

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The rate reduction models have been widely used to model the recurrent failure data for their capabilities in quantifying the repair effects. Despite the widespread popularity, there have been limited studies on statistical inference of most failure rate reduction models. In view of this fact, this study proposes a semiparametric estimation framework for a general class of such models, called extended geometric failure rate reduction (EGFRR) models. Covariates are considered in our analysis and their effects are modeled as a log-linear factor on the baseline failure rate. Unlike the existing inference methods for the EGFRR models that assume the failure data are censored at a fixed number of failures, our study considers covariates and time-censoring, which are more common in practice. The semiparametric maximum likelihood (ML) estimators are obtained by carefully constructing the likelihood function. Asymptotic properties including consistency and weak convergence of the ML estimators are established by using the properties of the martingale process. In addition, we show that the semiparametric estimators are asymptotically efficient. A real example from the automobile industry illustrates the usefulness of the proposed framework and extensive simulations show its outstanding performance when comparing with the existing methods.

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This study proposes a framework to analyze accelerated degradation testing (ADT) data in the presence of inspection effects. Motivated by a real dataset from the electric industry, we study two types of effects induced by inspections. After each inspection, the system degradation level instantaneously reduces by a random value. Meanwhile, the degrading rate is elevated afterwards. Considering the absence of observations due to practical reasons, we employ the expectation–maximization (EM) algorithm to analytically estimate the unknown parameters in a stepwise Wiener degradation process with covariates. Moreover, to maintain the level of generality for the adaption of the method in various scenarios, a confidence density approach is utilized to hierarchically estimate the parameters in the acceleration link function. The proposed methods can provide efficient parameter estimation under complex link functions with multiple stress factors. Further, confidence intervals are derived based on the large-sample approximation. Simulation studies and a case study from Schneider Electric are used to illustrate the proposed methods. The results show that the proposed model yields a remarkably better fit to the Schneider data in comparison to the conventional Wiener ADT model.

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Coronavirus disease-2019 (COVID-19) poses a significant threat to the population and urban sustainability worldwide. The surge mitigation is complicated and associates many factors, including the pandemic status, policy, socioeconomics and resident behaviours. Modelling and analytics with spatial-temporal big urban data are required to assist the mitigation of the pandemic. This study proposes a novel perspective to analyse the spatial-temporal potential exposure risk of residents by capturing human behaviours based on spatial-temporal car park availability data. Near real-time data from 1,904 residential car parks in Singapore, a classical megacity, are collected to analyse car mobility and its spatial-temporal heat map. The implementation of the circuit breaker, a COVID-19 measure, in Singapore has reduced the mobility and heat (daily frequency of mobility) significantly at about 30.0%. It contributes to a 44.3%–55.4% reduction in the transportation-related air emissions under two scenarios of travelling distance reductions. Urban sustainability impacts in both environment and economy are discussed. The spatial-temporal potential exposure risk mapping with space-time interactions is further investigated via an extended Bayesian spatial-temporal regression model. The maximal reduction rate of the defined potential exposure risk lowers to 37.6% by comparison with its peak value. The big data analytics of changes in car mobility behaviour and the resultant potential exposure risks can provide insights to assist in (a) designing a flexible circuit breaker exit strategy, (b) precise management via identifying and tracing hotspots on the mobility heat map, and (c) making timely decisions by fitting curves dynamically in different phases of COVID-19 mitigation. The proposed method has the potential to be used by decision-makers worldwide with available data to make flexible regulations and planning.

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The performance of units in the same batch can exhibit considerable heterogeneity due to the variation in the raw materials and fluctuation in the manufacturing process. For products suffering performance degradation in their use, such heterogeneity often results in an increase in the dispersion of the degradation paths of units in a population. The degradation rate of products can be unit-specific and often treated as random effects. This paper develops a novel random-effects Wiener process model to account for the unit-to-unit heterogeneity in the degradation, where the generalized inverse Gaussian (GIG) distribution is used to model the unit-specific degradation rate. The GIG distribution is a very general distribution with broad applications, which includes the inverse Gaussian (IG) distribution and the Gamma distribution as special cases. We investigate the model properties and develop an expectation maximization (EM) algorithm for parameter estimation. By comparing the proposed model with existing models on two real degradation datasets of the infrared LEDs and the GaAs lasers, we show that the proposed model is quite effective for degradation modeling with heterogeneous rates.

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This study proposes an efficient approach to detect one or more change points for gamma distribution. We plug a closed-form estimator into the gamma log-likelihood function to obtain a sharp approximation to the maximum of log-likelihood. We further derive a closed form calibration of approximate likelihood which is asymptotically equivalent to the exact log-likelihood. This circumvents iterative optimization procedures to find maximum likelihood estimates which can be a burden in detecting multiple change points. The simulation study shows that the approximation is accurate and the change points can be detected much faster. Two case studies on the time between events arising from industrial accidents are presented and extensively investigated.

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Dengue has been as an endemic with year-round presence in Singapore. In the recent years 2013, 2014, and 2016, there were several severe dengue outbreaks, posing serious threat to the public health. To proactively control and mitigate the disease spread, early warnings of dengue outbreaks, at which there are rapid and large-scale spread of dengue incidences, are extremely helpful. In this study, a two-step framework is proposed to predict dengue outbreaks and it is evaluated based on the dengue incidences in Singapore during 2012 to 2017. First, a generalized additive model (GAM) is trained based on the weekly dengue incidence data during 2006 to 2011. The proposed GAM is a one-week-ahead forecasting model, and it inherently accounts for the possible correlation among the historical incidence data, making the residuals approximately normally distributed. Then, an exponentially weighted moving average (EWMA) control chart is proposed to sequentially monitor the weekly residuals during 2012 to 2017. Our investigation shows that the proposed two-step framework is able to give persistent signals at the early stage of the outbreaks in 2013, 2014, and 2016, which provides early alerts of outbreaks and wins time for the early interventions and the preparation of necessary public health resources. In addition, extensive simulations show that the proposed method is comparable to other potential outbreak detection methods and it is robust to the underlying data-generating mechanisms.

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Many large organizations have developed ambitious programs to build reliability databases by collecting field failure data from a large variety of components. To make the database concise, only the number of component replacements in a component position during an operation time interval is reported in these databases. This leads to time-censoring in the aggregate failure data. Statistical inference for the time-censored aggregate data is challenging, because the likelihood function based on some common lifetime distributions can be intractable. In this study, we propose a general parametric estimation framework for the aggregate data. We first use the gamma distribution and the Inverse Gaussian (IG) distribution to model the aggregate data. Bayesian inference for the two models is discussed. Unlike the gamma/IG distribution, other lifetime distributions involve multiple integrals in the likelihood function, making the standard Bayesian inference difficult. To address the estimation problem, an approximate Bayesian computation algorithm that does not require evaluating the likelihood function is proposed, and its performance is assessed by simulation. As there are several candidate distributions, we further propose a model selection procedure to identify an appropriate distribution for the time-censored aggregate data. A real aggregate dataset extracted from a reliability database is used for illustration.

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The remaining useful lifetime (RUL) estimated from the in-situ degradation data has shown to be useful for online predictive maintenance. In the literature, the RUL is often estimated by assuming a soft-failure threshold for the degradation data. In practice, however, systems may not be subject to the degradation-induced soft failures. Instead, the systems are deemed to be fail when they cannot perform the intended function, and such failures are known as hard failures. Because there are no fixed thresholds for hard failures, the corresponding RUL estimation is not an easy task, which causes difficulties in finding the optimal maintenance schedule. In this study, a Weibull proportional hazards model is proposed to jointly model the degradation data and the failure time data. The degradation data are treated as the time-varying covariates so that the degradation does not directly lead to system failures, but increases the hazard rate of hard failures. A random-effects Wiener process is proposed to model the degradation data by considering the system heterogeneities. Based on the developed proportional hazards model, closed-form distribution of the RUL is derived upon each inspection and the optimal maintenance schedule is then obtained by minimizing the system maintenance cost. The proposed maintenance strategy is successfully applied to predictive maintenance of lead-acid batteries.

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In this study, we initiate a cyberinfrastructure solution by synergizing both the blockchain and Internet of Things (IoT) technologies for transportation insurance. The insurance premium related services are encapsulated in 'on-chain' chaincodes to perform over the facts on vehicle's trip and driver's behavior, which are deduced through 'off-chain' analytic services using the sensing data collected from vehicles' on-board sensors. A hybrid scheme coordinating both the permissioned (Hyperledger) and public (Ethereum) blockchains is proposed to exploit their respective capabilities in terms of high transaction throughput and built-in cryptocurrency. A working prototype platform is implemented with a basic premium calculation model. The prototype system is deployed across Amazon Web Services (AWSs) cloud in a real-world Internet environment. A comprehensive performance study from the aspects of throughput, latency, and resource usage under different configurations is presented to show the solution's feasibility. The design practice and research findings are concluded in consort with the experience gained for further enhancing the proposed solution and extending the functional features such as a more realistic insurance policy to be applied in generic vehicle insurance applications.

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Process capability indices (PCIs) are widely used to assess whether an in-control process meets manufacturing specifications. In most applications of classical PCIs, the process characteristic is assumed normally distributed. However, the normal distribution has been found inappropriate in various applications. In the literature, the percentile-based PCIs are widely used to deal with the nonnormal process. One problem associated with the percentile-based PCIs is that they do not provide a quantitative interpretation to the process capability. In this study, new PCIs that have a consistent quantification to the process capability for both normal and nonnormal processes are proposed. The proposed PCIs are generalizations of the classical normal PCIs in the sense that they are the same as the classical PCIs when the process characteristic follows a normal distribution, and they offer the same interpretation to the process capability as the classical PCIs when the process characteristic is nonnormal. We then discuss nonparametric and parametric estimation of the proposed PCIs. The nonparametric estimator is based on the kernel density estimation and confidence limits are obtained by the nonparametric bootstrap, while the parametric estimator is based on the maximum likelihood estimation and confidence limits are constructed by the method of generalized pivots. The proposed methodologies are demonstrated using a real example from a manufacturing factory.@en

Reliability data obtained from life tests and degradation tests have been extensively used for purposes such as estimating product reliability and predicting warranty costs. When there is more than one candidate model, an important task is to discriminate between the models. In the literature, the model discrimination was often treated as a hypothesis test and a pairwise model discrimination procedure was carried out. Because the null distribution of the test statistic is unavailable in most cases, the large sample approximation and the bootstrap were frequently used to find the acceptance region of the test. Although these two methods are asymptotically accurate, their performance in terms of size and power is not satisfactory in small sample size. To enhance the small-sample performance, we propose a new method to approximate the null distribution, which builds on the idea of generalized pivots. Conventionally, the generalized pivots were often used for interval estimation of a certain parameter or function of parameters in presence of nuisance parameters. In this study, we further extend the idea of generalized pivots to find the acceptance region of the model discrimination test. Through extensive simulations, we show that the proposed method performs better than the existing methods in discriminating between two lifetime distributions or two degradation models over a wide range of sample sizes. Two real examples are used to illustrate the proposed methods.

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Process capability indices (PCIs) are widely used to measure whether an in-control process conforms to manufacturing specifications. The normal distribution is assumed in most traditional applications of PCIs. Nevertheless, it is not uncommon that some quality characteristics have skewed distributions. In such cases, the gamma distribution is an appropriate model and percentile-based PCIs for the gamma process have been studied in the literature. In practical applications of PCIs, it is important to select an appropriate distribution between the normal and the gamma distributions based on historical data. In this study, we first construct a hypothesis test for model discrimination between the normal and the gamma distributions. Asymptotic distribution of the test statistic under the gamma process is derived. We then consider statistical inference for the percentile-based PCIs under the gamma process. The maximum likelihood method is used for point estimation and the method of generalized pivotal quantities is used for interval estimation. We demonstrate the proposed methods by a practical example.@en

Due to the variability of raw materials and the fluctuation in the manufacturing process, degradation of products may exhibit unit-to-unit variability in a population. The heterogeneous degradation rates can be viewed as random effects, which are often modeled by a normal distribution. Despite of its mathematical convenience, the normal distribution has certain limitations in modeling the random effects. In this study, we propose a novel random-effects Wiener process model based on ideas from accelerated failure time principle. An inverse Gaussian (IG) distribution can be used to characterize the unit-specific heterogeneity in degradation paths, which overcomes the disadvantages of the traditional models and provides more flexibility in the degradation modeling using Wiener processes. Properties of the model are investigated, and statistical inference based on the maximum likelihood estimation and the EM algorithm is established. An extension of the model to the constant-stress accelerated degradation test (ADT) is developed. The effectiveness and applicability of the proposed model are validated using a laser degradation dataset and an LED ADT dataset.

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Degradation data are an important source of product reliability information. Two popular stochastic models for degradation data are the Gamma process and the inverse Gaussian (IG) process, both of which possess monotone degradation paths. Although these two models have been used in numerous applications, the existing interval estimation methods are either inaccurate given a moderate sample size of the degradation data or require a significant computation time when the size of the degradation data is large. To bridge this gap, this article develops a general framework of interval estimation for the Gamma and IG processes based on the method of generalized pivotal quantities. Extensive simulations are conducted to compare the proposed methods with existing methods under moderate and large sample sizes. Degradation data from capacitors are used to illustrate the proposed methods.

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Field data provide important information about product quality and reliability. Many large organizations have developed ambitious reliability databases to trace field failure data of a variety of components on the systems they operate and maintain. Due to the exponential distribution assumption for the component lifetimes, the data in these databases are often aggregated. Specifically, individual lifetimes of the components are not available. Instead, each recorded data point is the cumulative operating time of one component position from system installation to the last component replacement, and the number of replacements in between. In the literature, the gamma distribution and the inverse Gaussian (IG) distribution have been used to fit the aggregate data, while the operating environment of different systems is often assumed the same. In order to capture possible heterogeneities among the systems, this study proposes the gamma random effects model and the IG random effects model. The expectation-maximization algorithm is used for point estimation of the parameters and an algorithm based on the generalized fiducial inference method is proposed for interval estimation. Simulation studies are conducted to assess the performance of the proposed inference methods. A real aggregate dataset is used for illustration.

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The generalized Pareto distribution (GPD) is widely used for extreme values over a threshold. Most existing methods for parameter estimation either perform unsatisfactorily when the shape parameter k is larger than 0.5, or they suffer from heavy computation as the sample size increases. In view of the fact that k > 0.5 is occasionally seen in numerous applications, including two illustrative examples used in this study, we remedy the deficiencies of existing methods by proposing two new estimators for the GPD parameters. The new estimators are inspired by the minimum distance estimation and the M-estimation in the linear regression. Through comprehensive simulation, the estimators are shown to perform well for all values of k under small and moderate sample sizes. They are comparable to the existing methods for k < 0.5 while perform much better for k > 0.5.

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