Recurrent events experienced by individual units or systems occur in many fields. The main target of this thesis is to develop formal tests for certain features of recurrent event processes, and to discuss their properties. In particular, carryover effects and time trends are considered. The former is related to clustering of events together in time, and the latter refers to a tendency for the rate of event occurrence to change over time in some systematic way. Score tests are developed for models incorporating carryover effects or time trends. The tests considered are easily interpreted and based on simple models but have good robustness properties against a range of carryover and trend alternatives. Asymptotic properties of test statistics are discussed when the number of processes approaches infinity as well as when one process is under observation for a long time. In applications involving multiple systems or individuals, heterogeneity is often apparent, and there is a need for tests developed for such cases. Allowance for heterogeneity is, therefore, considered. Methods are applied to data sets from industry and medicine. The results are supported by simulation studies.

Recurrent events experienced by individual units or systems occur in many fields. The main target of this thesis is to develop formal tests for certain features of recurrent event processes, and to discuss their properties. In particular, carryover effects and time trends are considered. The former is related to clustering of events together in time, and the latter refers to a tendency for the rate of event occurrence to change over time in some systematic way. Score tests are developed for models incorporating carryover effects or time trends. The tests considered are easily interpreted and based on simple models but have good robustness properties against a range of carryover and trend alternatives. Asymptotic properties of test statistics are discussed when the number of processes approaches infinity as well as when one process is under observation for a long time. In applications involving multiple systems or individuals, heterogeneity is often apparent, and there is a need for tests developed for such cases. Allowance for heterogeneity is, therefore, considered. Methods are applied to data sets from industry and medicine. The results are supported by simulation studies.

The behavior of repairable equipment is often modeled under assumptions such as perfect repair, minimal repair, or negligible repair. However the majority of equipment behavior does not fall into any of these categories. Rather, repair actions do take time and the condition of equipment following repair is not strictly "as good as new" or "as bad as it was" prior to repair. Non-homogeneous processes that reflect this type of behavior are not studied nearly as much as the minimal repair case, but they far more realistic in many situations. For this reason, the quasi-renewal process provides an appealing alternative to many existing models for describing a non-homogeneous process. A quasi-renewal process is characterized by a parameter that indicates process deterioration or improvement by falling in the interval [0,1) or (1,Infinity) respectively. This parameter is the amount by which subsequent operation or repair intervals are scaled in terms of the immediately previous operation or repair interval. Two equivalent expressions for the point availability of a system with operation intervals and repair intervals that deteriorate according to a quasi-renewal process are constructed. In addition to general expressions for the point availability, several theoretical distributions on the operation and repair intervals are considered and specific forms of the quasi-renewal and point availability functions are developed. The two point availability expressions are used to provide upper and lower bounds on the approximated point availability. Numerical results and general behavior of the point availability and quasi-renewal functions are examined. The framework provided here allows for the description and prediction of the time-dependent behavior of a non-homogeneous process without the assumption of limiting behavior, a specific cost structure, or minimal repair. / Ph. D.

Os modelos de sistemas reparáveis usuais são os de reparo mínimo, perfeito e imperfeito, sendo que, na literatura, o modelo de reparo mínimo é o mais explorado. Em sistemas reparáveis é comum que componentes do mesmo tipo sejam estudados e nestes casos é relevante verificar a heterogeneidade entre eles. Segundo Vaupel et al. (1979), os métodos padrões em análise de dados de sistemas reparáveis ignoram a heterogeneidade não observada e em alguns casos esta deveria ser considerada. Tal variabilidade pode ser estimada a partir dos modelos de fragilidade, caracterizados pela utilização de um efeito aleatório. Propõe-se o modelo de reparo mínimo com fragilidade, afim de estimar a heterogeneidade não observada entre sistemas. Para este modelo foi realizado um estudo e simulação como objetivo de analisar as propriedades frequentistas do processo de estimação. A aplicação em um conjunto de dados reais mostrou a aplicabilidade do modelo proposto, em que a estimação dos parâmetros foram determinadas a partir das abordagens de máxima verossimilhança e Bayesiana. / The usual models in repair able systems are minimal, perfect and imperfect repair, and, in the literature, the minimum repair model is the most explored. In repair able systems it is common that the same type of components are studied and in these cases is relevant to verify the heterogeneity between them. According to Vaupel et al. (1979), the standard methods for analysis of repair able systems data ignore the heterogeneity not observed and in some cases this should be considered. Such variability can be estimated from frailty models, characterized by using a random effect. It is proposed that the minimum repair model with frailty in order to estimate the heterogeneity not observed between systems. For this model it was conducted a simulation study in order to analyze the frequentist properties of the estimation process. The application of a real data set showed the applicability of the proposed model, in which the estimation of the parameters were determined from maximum likelihood and Bayesian approaches.

Submitted by Daniele Amaral (daniee_ni@hotmail.com) on 2016-09-15T15:15:11Z No. of bitstreams: 1 DissCCF.pdf: 1472861 bytes, checksum: 937810926b98a08677085a559d13aec9 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-16T19:51:30Z (GMT) No. of bitstreams: 1 DissCCF.pdf: 1472861 bytes, checksum: 937810926b98a08677085a559d13aec9 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-16T19:51:36Z (GMT) No. of bitstreams: 1 DissCCF.pdf: 1472861 bytes, checksum: 937810926b98a08677085a559d13aec9 (MD5) / Made available in DSpace on 2016-09-16T19:51:41Z (GMT). No. of bitstreams: 1 DissCCF.pdf: 1472861 bytes, checksum: 937810926b98a08677085a559d13aec9 (MD5) Previous issue date: 2015-09-15 / Não recebi financiamento / The usual models in repairable systems are minimal, perfect and imperfect repair, and, in the literature, the minimum repair model is the most explored. In repairable systems it is common that the same type of components are studied and in these cases is relevant to verify the heterogeneity between them. According to Vaupel et al. (1979), the standard methods for analysis of repairable systems data ignore the heterogeneity not observed and in some cases this should be considered. Such variability can be estimated from frailty models, characterized by using a random e ect. It is proposed that the minimum repair model with frailty in order to estimate the heterogeneity not observed between systems. For this model it was conducted a simulation study in order to analyze the frequentist properties of the estimation process. The application of a real data set showed the applicability of the proposed model, in which the estimation of the parameters were determined from maximum likelihood and Bayesian approaches. / Os modelos de sistemas reparáveis usuais são os de reparo mí nimo, perfeito e imperfeito, sendo que, na literatura, o modelo de reparo mí nimo e o mais explorado. Em sistemas reparáveis e comum que componentes do mesmo tipo sejam estudados e nestes casos é relevante verifi car a heterogeneidade entre eles. Segundo Vaupel et al. (1979), os métodos padrões em análise de dados de sistemas reparáveis ignoram a heterogeneidade não observada e em alguns casos esta deveria ser considerada. Tal variabilidade pode ser estimada a partir dos modelos de fragilidade, caracterizados pela utilização de um efeito aleat ório. Propõe-se o modelo de reparo mí nimo com fragilidade, a fi m de estimar a heterogeneidade não observada entre sistemas. Para este modelo foi realizado um estudo de simula ção com o objetivo de analisar as propriedades frequentistas do processo de estimação. A aplicação em um conjunto de dados reais mostrou a aplicabilidade do modelo proposto, em que a estima ção dos parâmetros foram determinadas a partir das abordagens de m áxima verossimilhan ça e Bayesiana.

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2016-08-05T13:45:36Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) 2016 02 Dissertação V18.pdf: 847550 bytes, checksum: b3b34305dd3687f0d1a2e2aa1a8ef506 (MD5) / Made available in DSpace on 2016-08-05T13:45:36Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) 2016 02 Dissertação V18.pdf: 847550 bytes, checksum: b3b34305dd3687f0d1a2e2aa1a8ef506 (MD5) Previous issue date: 2016-02-19 / CAPEs / A terceirização da manutenção e a adesão da garantia estendida para equipamentos hospitalares tem se tornado uma tendência ao longo das últimas décadas, pois estão relacionadas com a crescente complexidade e modernização dos dispositivos médicos, bem como, com a recorrente prática da exclusividade do fabricante na realização da manutenção. Geralmente, o relacionamento entre a instituição de saúde e o fabricante é conduzido por meio de um documento, especificando questões como: nível de confiabilidade, desempenho operacional, disponibilidade, duração da garantia, preço da manutenção, penalidades e a política de manutenção a ser implementada. Sendo assim, a presente dissertação estuda quantitativamente o problema de garantia estendida para equipamentos hospitalares, por meio da junção de duas ferramentas: o jogo de Stackelberg, designado para estruturar a forma de relacionamento entre as empresas, e o Processo de Renovação Generalizado, responsável processo de falha – reparo do equipamento (reparo imperfeito). Um cenário foi criado para a aplicação de tais métodos. Inicialmente, o fabricante ao vender um equipamento hospitalar também oferece duas possibilidades para a execução da manutenção: a primeira, garantia estendida, e a segunda, serviço sob demanda. Posteriormente, a decisão do hospital é influenciada pela estrutura de preços imposta do fabricante, a confiabilidade do equipamento e o seu grau de aversão ao risco, visto que as falhas do dispositivo são eventos aleatórios. Para ilustrar tal situação, realiza-se um exemplo numérico com dados de falha e reparo de um Angiográfo. O equilíbrio do modelo implica na maximização do lucro esperado do fabricante e o hospital decidindo pela adesão da garantia estendida. Adicionalmente, comparando as soluções do reparo imperfeito com os cenários de reparos perfeito e mínimo, observou-se similaridade nas estratégias para os casos de reparos imperfeito e perfeito, enquanto que, na relação entre os reparos imperfeito e mínimo as estratégias darse- ão de maneira oposta. Finalmente, o lucro esperado do fabricante diminui conforme aumenta o número médio de falhas. / The outsourcing of maintenance and the acquisition of extended warranty for hospital equipment has become a trend over the past few decades, since they are related to the growing complexity, modernization of medical devices and the recurring practice of the manufacturer's exclusivity in performing maintenance services. Generally, the relationship between the health institution and the manufacturer is conducted through a document specifying the following issues: level of reliability, operating performance, availability, warranty period, maintenance price, penalties and maintenance policy to be implemented. Under these circumstances, this thesis analyzes the problem of the extended warranty for clinical equipment by joining two tools: the Stackelberg game, designed to model the relation between companies and the Generalized Renewal Process, employed for modeling failure-repair process (imperfect repair). A scenario was created for the application of such methods. Initially, the manufacturer intends to sell a medical equipment and also offers two maintenance possibilities: first, an extended warranty, and second, maintenance services on demand. Subsequently, the hospital's decision is influenced by manufacturer's price structure, equipment reliability and the degree of risk aversion, since the failures occurrences of the device are random events. To illustrate this situation, an application example with failure and repair data of an Angiography equipment is presented. The equilibrium of model implies in expected profit maximization for the manufacturer and hospital chooses the extended warranty option. In addition, by comparing the solutions of the imperfect repair with perfect and minimal repair scenarios, a similarity in the strategies adopted in cases of imperfect and perfect repairs was observed, while the strategies were opposite when comparing imperfect and minimal repairs. Finally, the expected profit of the manufacturer decreases as the average number of failures increases.

Dans le contexte de la modélisation aléatoire des événements récurrents, un modèle statistique particulier est exploré. Ce modèle est fondé sur la théorie des processus de comptage et est construit dans le cadre d'analyse de défaillances dans les réseaux d'eau. Dans ce domaine nous disposons de données sur de nombreux systèmes observés durant une certaine période de temps. Les systèmes étant posés à des instants différents, leur âge est utilisé en tant qu'échelle temporelle dans la modélisation. Le modèle tient compte de l'historique incomplet d'événements, du vieillissement des systèmes, de l'impact négatif des défaillances précédentes sur l'état des systèmes et des covariables. Le modèle est positionné parmi d'autres approches visant à l'analyse d'événements récurrents utilisées en biostatistique et en fiabilité. Les paramètres du modèle sont estimés par la méthode du Maximum de Vraisemblance (MV). Une covariable dépendante du temps est intégrée au modèle. Il est supposé qu'elle est extérieure au processus de défaillance et constante par morceaux. Des méthodes heuristiques sont proposées afin de tenir compte de cette covariable lorsqu'elle n'est pas observée. Des méthodes de simulation de données artificielles et des estimations en présence de la covariable temporelle sont proposées. Les propriétés de l'estimateur (la normalité, le biais, la variance) sont étudiées empiriquement par la méthode de Monte Carlo. L'accent est mis sur la présence de deux directions asymptotiques : asymptotique en nombre de systèmes n et asymptotique en durée d'observation T. Le comportement asymptotique de l'estimateur MV constaté empiriquement est conforme aux résultats théoriques classiques. Il s'agit de l'asymptotique en n. Le comportement T-asymptotique constaté empiriquement n'est pas classique. L'analyse montre également que les deux directions asymptotiques n et T peuvent être combinées en une unique direction : le nombre d'événements observés. Cela concerne les paramètres classiques du modèle (les coefficients associés aux covariables fixes et le paramètre caractérisant le vieillissement des systèmes). Ce n'est en revanche pas le cas pour le coefficient associé à la covariable temporelle et pour le paramètre caractérisant l'impact négatif des défaillances précédentes sur le comportement futur du système. La méthodologie développée est appliquée à l'analyse des défaillances des réseaux d'eau. L'influence des variations climatiques sur l'intensité de défaillance est prise en compte par une covariable dépendante du temps. Les résultats montrent globalement une amélioration des prédictions du comportement futur du processus lorsque la covariable temporelle est incluse dans le modèle. / In the context of stochastic modeling of recurrent events, a particular model is explored. This model is based on the counting process theory and is built to analyze failures in water distribution networks. In this domain the data on a large number of systems observed during a certain time period are available. Since the systems are installed at different dates, their age is used as a time scale in modeling. The model accounts for incomplete event history, aging of systems, negative impact of previous failures on the state of systems and for covariates.The model is situated among other approaches to analyze the recurrent events, used in biostatistics and in reliability. The model parameters are estimated by the Maximum Likelihood method (ML). A method to integrate a time-dependent covariate into the model is developed. The time-dependent covariate is assumed to be external to the failure process and to be piecewise constant. Heuristic methods are proposed to account for influence of this covariate when it is not observed. Methods for data simulation and for estimations in presence of the time-dependent covariate are proposed. A Monte Carlo study is carried out to empirically assess the ML estimator's properties (normality, bias, variance). The study is focused on the doubly-asymptotic nature of data: asymptotic in terms of the number of systems n and in terms of the duration of observation T. The asymptotic behavior of the ML estimator, assessed empirically agrees with the classical theoretical results for n-asymptotic behavior. The T-asymptotics appears to be less typical. It is also revealed that the two asymptotic directions, n and T can be combined into one unique direction: the number of observed events. This concerns the classical model parameters (the coefficients associated to fixed covariates, the parameter characterizing aging of systems). The presence of one unique asymptotic direction is not obvious for the time-dependent covariate coefficient and for a parameter characterizing the negative impact of previous events on the future behavior of a system.The developed methodology is applied to the analysis of failures of water networks. The influence of climatic variations on failure intensity is assessed by a time-dependent covariate. The results show a global improvement in predictions of future behavior of the process when the time-dependent covariate is included into the model.

We consider a class of periodic preventive maintenance (PM) optimization problems, for a single piece of equipment that deteriorates with time or use, and can be repaired upon failure, through corrective maintenance (CM). We develop analytical and simulation-based optimization models that seek an optimal periodic PM policy, which minimizes the sum of the expected total cost of PMs and the risk-averse cost of CMs, over a finite planning horizon. In the simulation-based models, we assume that both types of maintenance actions are imperfect, whereas our analytical models consider imperfect PMs with minimal CMs. The effectiveness of maintenance actions is modeled using age reduction factors. For a repairable unit of equipment, its virtual age, and not its calendar age, determines the associated failure rate. Therefore, two sets of parameters, one describing the effectiveness of maintenance actions, and the other that defines the underlying failure rate of a piece of equipment, are critical to our models. Under a given maintenance policy, the two sets of parameters and a virtual-age-based age-reduction model, completely define the failure process of a piece of equipment. In practice, the true failure rate, and exact quality of the maintenance actions, cannot be determined, and are often estimated from the equipment failure history. We use a Bayesian approach to parameter estimation, under which a random-walk-based Gibbs sampler provides posterior estimates for the parameters of interest. Our posterior estimates for a few datasets from the literature, are consistent with published results. Furthermore, our computational results successfully demonstrate that our Gibbs sampler is arguably the obvious choice over a general rejection sampling-based parameter estimation method, for this class of problems. We present a general simulation-based periodic PM optimization model, which uses the posterior estimates to simulate the number of operational equipment failures, under a given periodic PM policy. Optimal periodic PM policies, under the classical maximum likelihood (ML) and Bayesian estimates are obtained for a few datasets. Limitations of the ML approach are revealed for a dataset from the literature, in which the use of ML estimates of the parameters, in the maintenance optimization model, fails to capture a trivial optimal PM policy. Finally, we introduce a single-stage and a two-stage formulation of the risk-averse periodic PM optimization model, with imperfect PMs and minimal CMs. Such models apply to a class of complex equipment with many parts, operational failures of which are addressed by replacing or repairing a few parts, thereby not affecting the failure rate of the equipment under consideration. For general values of PM age reduction factors, we provide sufficient conditions to establish the convexity of the first and second moments of the number of failures, and the risk-averse expected total maintenance cost, over a finite planning horizon. For increasing Weibull rates and a general class of increasing and convex failure rates, we show that these convexity results are independent of the PM age reduction factors. In general, the optimal periodic PM policy under the single-stage model is no better than the optimal two-stage policy. But if PMs are assumed perfect, then we establish that the single-stage and the two-stage optimization models are equivalent. / text