Parametric inference for multiple repairable systems under dependent competing risks

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/115899/1/asmb2079.pd

Inferences on the power-law process with applications to repairable systems

System testing is very time-consuming and costly, especially for complex high-cost and high-reliability systems. For this reason, the number of failures needed for the developmental phase of system testing should be relatively small in general. To assess the reliability growth of a repairable system, the generalized confidence interval and the modified signed log-likelihood ratio test for the scale parameter of the power-law process are studied concerning incomplete failure data. Specifically, some recorded failure times in the early developmental phase of system testing cannot be observed; this circumstance is essential to establish a warranty period or determine a maintenance phase for repairable systems. For the proposed generalized confidence interval, we have found that this method is not biased estimates which can be seen from the coverage probabilities obtained from this method being close to the nominal level 0.95 for all levels of γ and β. When the performance of the proposed method and the existing method are compared and validated regarding average widths, the simulation results show that the proposed method is superior to another method due to shorter average widths when the predetermined number of failures is small. For the proposed modified signed log-likelihood ratio test, we have found that this test performs well in controlling type I errors for complete failure data, and it has desirable powers for all parameters configurations even for the small number of failures. For incomplete failure data, the proposed modified signed log-likelihood ratio test is preferable to the signed log-likelihood ratio test in most situations in terms of controlling type I errors. Moreover, the proposed test also performs well when the missing ratio is up to 30% and n \u3e 10. In terms of empirical powers, the proposed modified signed log-likelihood ratio test is superior to another test for most situations. In conclusion, it is quite clear that the proposed methods, the generalized confidence interval, and the modified signed log-likelihood ratio test, are practically useful to save business costs and time during the developmental phase of system testing since the only small number of failures is required to test systems, and it yields precise results

Nonparametric estimation of first passage time distributions in flowgraph models

Statistical flowgraphs represent multistate semi-Markov processes using integral transforms of transition time distributions between adjacent states; these are combined algebraically and inverted to derive parametric estimates for first passage time distributions between nonadjacent states. This dissertation extends previous work in the field by developing estimation methods for flowgraphs using empirical transforms based on sample data, with no assumption of specific parametric probability models for transition times. We prove strong convergence of empirical flowgraph results to the exact parametric results; develop alternatives for numerical inversion of empirical transforms and compare them in terms of computational complexity, accuracy, and ability to determine error bounds; discuss (with examples) the difficulties of determining confidence bands for distribution estimates obtained in this way; develop confidence intervals for moment-based quantities such as the mean; and show how methods based on empirical transforms can be modified to accommodate censored data. Several applications of the nonparametric method, based on reliability and survival data, are presented in detail

On the Statistical Modeling and Analysis of Repairable Systems

We review basic modeling approaches for failure and maintenance data from repairable systems. In particular we consider imperfect repair models, defined in terms of virtual age processes, and the trend-renewal process which extends the nonhomogeneous Poisson process and the renewal process. In the case where several systems of the same kind are observed, we show how observed covariates and unobserved heterogeneity can be included in the models. We also consider various approaches to trend testing. Modern reliability data bases usually contain information on the type of failure, the type of maintenance and so forth in addition to the failure times themselves. Basing our work on recent literature we present a framework where the observed events are modeled as marked point processes, with marks labeling the types of events. Throughout the paper the emphasis is more on modeling than on statistical inference.Comment: Published at http://dx.doi.org/10.1214/088342306000000448 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Contributions to Reliability and Lifetime Data Analysis.

This dissertation deals with problems in reliability and lifetime data analysis. The first part focuses on the study of graphical estimators from probability plots with right censored data. The second part deals with reliability inference for repairable systems. Probability plots are popular graphical tools for assessing parametric distributional assumptions among reliability engineers and other practitioners. They are particularly well suited for location-scale families or those that can be transformed to such families. When the plot indicates a reasonable conformity to the assumed family, it is common to estimate the underlying location and scale parameters by fitting a line through the plot. This quick-and-easy method is especially useful with censored data. Indeed, the current version of a popular statistical software package uses this as the default estimation method. Part I of the dissertation investigates the properties of graphical estimators with multiply right-censored data and compares their performance to maximum likelihood estimators. Large-sample results on consistency, asymptotic normality, and asymptotic variance expressions are obtained. Small-sample properties are studied through simulation for selected distributions and censoring patterns. The results presented in this study extend the work of Nair (1984) to right-censored data. Analysis of failure data arising from repairable systems has received considerable attention in the statistical, engineering, computer software, and medical literature. Data pertaining to a repairable system is viewed as some type of `recurrent event'. Part II of the dissertation investigates some models and methodologies for analyzing failures from repairable systems with multiple failure modes. We consider the case where the cause-specific failures (from each failure mode) follow some counting processes with an emphasis on nonhomogeneous Poisson processes (NHPPs). Some properties of the data are characterized and estimation methods are studied, both from a single system and multiple systems assuming independence of the failure modes. Some results are also developed when there is partial masking of the failure modes. The NHPP case with a power law intensity function is studied in detail.Ph.D.StatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/57718/2/asomboon_1.pd

Statistical Reliability with Applications

This chapter reviews fundamental ideas in reliability theory and inference. The first part of the chapter accounts for lifetime distributions that are used in engineering reliability analyis, including general properties of reliability distributions that pertain to lifetime for manufactured products. Certain distributions are formulated on the basis of simple physical properties, and other are more or less empirical. The first part of the chapter ends with a description of graphical and analytical methods to find appropriate lifetime distributions for a set of failure data. The second part of the chapter describes statistical methods for analyzing reliability data, including maximum likelihood estimation and likelihood ratio testing. Degradation data are more prevalent in experiments in which failure is rare and test time is limited. Special regression techniques for degradation data can be used to draw inference on the underlying lifetime distribution, even if failures are rarely observed. The last part of the chapter discusses reliability for systems. Along with the components that comprise the system, reliability analysis must take account of the system configuration and (stochastic) component dependencies. System reliability is illustrated with an analysis of logistics systems (e.g., moving goods in a system of product sources and retail outlets). Robust reliability design can be used to construct a supply chain that runs with maximum efficiency or minimum cost

Bi-Directional Testing for Change Point Detection in Poisson Processes

Point processes often serve as a natural language to chronicle an event\u27s temporal evolution, and significant changes in the flow, synonymous with non-stationarity, are usually triggered by assignable and frequently preventable causes, often heralding devastating ramifications. Examples include amplified restlessness of a volcano, increased frequencies of airplane crashes, hurricanes, mining mishaps, among others. Guessing these time points of changes, therefore, merits utmost care. Switching the way time traditionally propagates, we posit a new genre of bidirectional tests which, despite a frugal construct, prove to be exceedingly efficient in culling out non-stationarity under a wide spectrum of environments. A journey surveying a lavish class of intensities, ranging from the tralatitious power laws to the deucedly germane rough steps, tracks the established unidirectional forward and backward test\u27s evolution into a p-value induced dual bidirectional test, the best member of the proffered category. Niched within a hospitable Poissonian framework, this dissertation, through a prudent harnessing of the bidirectional category\u27s classification prowess, incites a refreshing alternative to estimating changes plaguing a soporific flow, by conducting a sequence of tests. Validation tools, predominantly graphical, rid the structure of forbidding technicalities, aggrandizing the swath of applicability. Extensive simulations, conducted especially under hostile premises of hard non-stationarity detection, document minimal estimation error and reveal the algorithm\u27s obstinate versatility at its most unerring

Étude de la réhabilitation sismique d'un pont avec des isolateurs en caoutchouc à basse température par le biais de surfaces de fragilité

Abstract : In Quebec, Canada, due to aging and deficient seismic detailing, bridges are susceptible to important damage in the occurrence of a strong earthquake. To enhance the seismic performance of the provincial bridge inventory, the replacement of typical bearings with natural rubber isolators has shown to be a potentially efficient retrofitting measure. However, variations in the mechanical properties of the isolators due to environmental conditions can affect the seismic performance. For instance, rubber undergoes substantial stiffening when exposed to low temperatures, as those typically observed during winter in eastern Canada. In bridge-type structures, the thermal stiffening of isolators increases the forces transmitted to the substructure, which in turn becomes more prone to damage. A more detailed consideration of the thermal effects on the seismic performance of typical provincial bridges is thus necessary. In this study, fragility surfaces are used to assess the vulnerability of a typical bridge in Quebec when retrofitted with natural rubber isolators under the concomitant actions of earthquakes and low temperatures. Bridges are composed of several different components with distinguished behaviors and complex interactions under seismic excitation. Owing to the importance of the contribution of different components to the bridge fragility, the first part of this study focuses on the construction of multivariate probabilistic seismic demand models (PSDM). The validity of the commonly adopted assumptions has been criticized and their impact on fragility estimates is not fully understood. A multivariate PSDM approach is thus developed coupling the multiple-stripe analysis and Gaussian mixture models. The novel approach concomitantly captures the complexity of the dynamic response of multicomponent structures and models their uncertainties and correlation. The proposed approach is then used to assess the potential bias introduced by poor modeling on fragility and risk estimates of a real as-built case-study bridge. This PSDM strategy then is adopted to translate the uncertainty and the correlation of the response of the case-study bridge components when retrofitted. Fragility surfaces based on logistic regression depict the effects of thermal stiffening of isolators on the performance of the bridge in both component- and system-level. A beneficial combination is revealed between the decoupling effect provided by isolators and the lateral restraining action of the abutment wing walls depending on the provide clearances. The derivation of fragility surfaces for isolated bridges in cold regions sheds new light on the challenges of retrofitting structures exposed to multiple extreme environments (e.g., seismic and thermal). Overall the presented results can facilitate seismic vulnerability modeling and retrofit assessment of these complex systems and afford valuable practical impacts. The insights and methodological advances can prompt research and applications well beyond the case study structures considered in the thesis, and have broad impacts.Au Québec, Canada, en raison du vieillissement et de détails insuffisants de dimensionnement sismique, les ponts sont susceptibles de subir des dommages importants en cas de fort séisme. Pour améliorer la performance sismique de l'inventaire des ponts de la province, le remplacement des appareils d'appui classiques par des isolateurs en caoutchouc naturel s'est avéré une mesure de réhabilitation potentiellement efficace. Cependant, les variations des propriétés mécaniques des isolateurs dues aux conditions environnementales peuvent affecter la performance sismique. Par exemple, le caoutchouc subit un raidissement important lorsqu'il est exposé aux basses températures, comme celles typiquement observées pendant les hivers dans l'est du Canada. Dans les ponts, le raidissement thermique des isolateurs augmente les forces transmises à la sous-structure, qui devient alors plus susceptible d'être endommagée. Une étude plus détaillée des effets thermiques sur la performance sismique des ponts provinciaux typiques est donc nécessaire. Des surfaces de fragilité sont donc utilisées pour évaluer la vulnérabilité d'un pont typique au Québec réhabilité avec des isolateurs en caoutchouc naturel sous les actions concomitantes des séismes et des basses températures. Les ponts sont composés de plusieurs éléments différents ayant des comportements distincts et des interactions complexes sous une excitation sismique. En raison de l'importance de la contribution de plusieurs composants à la fragilité du pont, la première partie de cette étude se concentre sur la construction de modèles probabilistes multivariés de demande sismique (PSDM). On a critiqué la validité des hypothèses couramment adoptées et leur impact sur les estimations de fragilité n'est pas entièrement compris. Une approche PSDM multivariée est donc développée en couplant l'analyse de bandes multiples et les modèles de mélange gaussien. La nouvelle approche capture de manière concomitante la complexité de la réponse dynamique et modélise les incertitudes et la corrélation. On évalue ensuite le biais potentiel introduit par une mauvaise modélisation sur les estimations de fragilité et de risque d'un pont réel tel que construit. Cette stratégie PSDM est ensuite adoptée pour traduire la réponse des composants du pont de l'étude de cas lorsqu'il est réhabilité. Les surfaces de fragilité basées sur la régression logistique décrivent les effets du raidissement thermique des isolateurs sur les performances du pont, tant au niveau des composants que du système. Une combinaison bénéfique est révélée entre l'effet de découplage des isolateurs et l'action de retenue latérale des murs en fonction des écarts fournis. La dérivation des surfaces de fragilité pour les ponts isolés dans les régions froides jette un nouvel éclairage sur les défis de la réhabilitation des structures exposées à de multiples environnements extrêmes (sismiques et thermiques). Dans l'ensemble, les résultats présentés peuvent faciliter la modélisation de la vulnérabilité sismique et l'évaluation de la réhabilitation de ces systèmes complexes et avoir des répercussions pratiques importantes. Les idées et les avancées méthodologiques peuvent susciter des recherches et des applications bien au-delà des structures étudiées dans la thèse, et en avoir un large impact