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.

This research has developed several models and methodologies with the aim of improving the accuracy and applicability of reliability predictions for complex repairable systems. A repairable system is usually defined as one that will be repaired to recover its functions after each failure. Physical assets such as machines, buildings, vehicles are often repairable. Optimal maintenance strategies require the prediction of the reliability of complex repairable systems accurately. Numerous models and methods have been developed for predicting system reliability. After an extensive literature review, several limitations in the existing research and needs for future research have been identified. These include the follows: the need for an effective method to predict the reliability of an asset with multiple preventive maintenance intervals during its entire life span; the need for considering interactions among failures of components in a system; and the need for an effective method for predicting reliability with sparse or zero failure data. In this research, the Split System Approach (SSA), an Analytical Model for Interactive Failures (AMIF), the Extended SSA (ESSA) and the Proportional Covariate Model (PCM), were developed by the candidate to meet the needs identified previously, in an effective manner. These new methodologies/models are expected to rectify the identified limitations of current models and significantly improve the accuracy of the reliability prediction of existing models for repairable systems. The characteristics of the reliability of a system will alter after regular preventive maintenance. This alternation makes prediction of the reliability of complex repairable systems difficult, especially when the prediction covers a number of imperfect preventive maintenance actions over multiple intervals during the asset's lifetime. The SSA uses a new concept to address this issue effectively and splits a system into repaired and unrepaired parts virtually. SSA has been used to analyse system reliability at the component level and to address different states of a repairable system after single or multiple preventive maintenance activities over multiple intervals. The results obtained from this investigation demonstrate that SSA has an excellent ability to support the making of optimal asset preventive maintenance decisions over its whole life. It is noted that SSA, like most existing models, is based on the assumption that failures are independent of each other. This assumption is often unrealistic in industrial circumstances and may lead to unacceptable prediction errors. To ensure the accuracy of reliability prediction, interactive failures were considered. The concept of interactive failure presented in this thesis is a new variant of the definition of failure. The candidate has made several original contributions such as introducing and defining related concepts and terminologies, developing a model to analyse interactive failures quantitatively and revealing that interactive failure can be either stable or unstable. The research results effectively assist in avoiding unstable interactive relationship in machinery during its design phase. This research on interactive failures pioneers a new area of reliability prediction and enables the estimation of failure probabilities more precisely. ESSA was developed through an integration of SSA and AMIF. ESSA is the first effective method to address the reliability prediction of systems with interactive failures and with multiple preventive maintenance actions over multiple intervals. It enhances the capability of SSA and AMIF. PCM was developed to further enhance the capability of the above methodologies/models. It addresses the issue of reliability prediction using both failure data and condition data. The philosophy and procedure of PCM are different from existing models such as the Proportional Hazard Model (PHM). PCM has been used successfully to investigate the hazard of gearboxes and truck engines. The candidate demonstrated that PCM had several unique features: 1) it automatically tracks the changing characteristics of the hazard of a system using symptom indicators; 2) it estimates the hazard of a system using symptom indicators without historical failure data; 3) it reduces the influence of fluctuations in condition monitoring data on hazard estimation. These newly developed methodologies/models have been verified using simulations, industrial case studies and laboratory experiments. The research outcomes of this research are expected to enrich the body of knowledge in reliability prediction through effectively addressing some limitations of existing models and exploring the area of interactive failures.

Les travaux présentés dans cette thèse concernent l'analyse statistique de cas particuliers du processus de Cox. Dans une première partie, nous proposons une synthèse des résultats existants sur le processus power-law (processus d'intensité puissance), synthèse qui ne peut être exhaustive étant donné la popularité de ce processus. Nous considérons une approche bayésienne pour l'inférence des paramètres de ce processus qui nous conduit à introduire et à étudier en détails une distribution que nous appelons loi H-B. Cette loi est une loi conjuguée. Nous proposons des stratégies d'élicitation des hyperparamètres et étudions le comportement des estimateurs de Bayes par des simulations. Dans un deuxième temps, nous étendons ces travaux au cas du processus d’intensité exponentielle (exponential-law process). De la même façon, nous définissons et étudions une loi conjuguée pour l'analyse bayésienne de ce dernier. Dans la dernière partie de la thèse, nous considérons un processus auto-excité qui intègre une covariable. Ce travail est motivé, à l'origine, par un problème de fiabilité qui concerne des données de défaillances de matériels exposés à des environnements sévères. Les résultats sont illustrés par des applications sur des données d'activités orageuses collectées dans deux départements français. Enfin, nous donnons quelques directions de travail et perspectives de futurs développements de l'ensemble de nos travaux. / The work presented in this PhD dissertation concerns the statistical analysis of some particular cases of the Cox process. In a first part, we study the power-law process (PLP). Since the literature for the PLP is abundant, we suggest a state-of-art for the process. We consider the classical approach and recall some important properties of the maximum likelihood estimators. Then we investigate a Bayesian approach with noninformative priors and conjugate priors considering different parametrizations and scenarios of prior guesses. That leads us to define a family of distributions that we name H-B distribution as the natural conjugate priors for the PLP. Bayesian analysis with the conjugate priors are conducted via a simulation study and an application on real data. In a second part, we study the exponential-law process (ELP). We review the maximum likelihood techniques. For Bayesian analysis of the ELP, we define conjugate priors: the modified- Gumbel distribution and Gamma-modified-Gumbel distribution. We conduct a simulation study to compare maximum likelihood estimates and Bayesian estimates. In the third part, we investigate self-exciting point processes and we integrate a power-law covariate model to this intensity of this process. A maximum likelihood procedure for the model is proposed and the Bayesian approach is suggested. Lastly, we present an application on thunderstorm data collected in two French regions. We consider a strategy to define a thunderstorm as a temporal process associated with the charges in a particular location. Some selected thunderstorms are analyzed. We propose a reduced maximum likelihood procedure to estimate the parameters of the Hawkes process. Then we fit some thunderstorms to the power-law covariate self-exciting point process taking into account the associated charges. In conclusion, we give some perspectives for further work.