Cette thèse aborde deux sujets peu traités dans la littérature concernant le théorie des valeurs extrêmes : celui des observations en présence de covariables et celui des mesures de dépendance pour des paires d'observations. Dans la première partie de cette thèse, nous avons considéré le cas où la variable d'intérêt est observée simultanément avec une covariable, pouvant être fixe ou aléatoire. Dans ce contexte, l'indice de queue dépend de la covariable et nous avons proposé des estimateurs de ce paramètre dont nous avons étudié les propriétés asymptotiques. Leurs comportements à distance finie ont été validés par simulations. Puis, dans la deuxième partie, nous nous sommes intéressés aux extrêmes multivariés et plus particulièrement à mesurer la dépendance entre les extrêmes. Dans une situation proche de l'indépendance asymptotique, il est très difficile de mesurer cette dépendance et de nouveaux modèles doivent être introduits. Dans ce contexte, nous avons adapté un outil de géostatistique, le madogramme, et nous avons étudié ses propriétés asymptotiques. Ses performances sur simulations et données réelles ont également été exhibées. Cette thèse offre de nombreuses perspectives, tant sur le plan pratique que théorique dont une liste non exhaustive est présentée en conclusion de la thèse. / This thesis presents a study of the extreme value theory and is focused on two subjects rarely analyzed: observations associated with covariates and dependence measures for pairs of observations.In the first part, we considered the case where the variable of interest is simultaneously recorded with a covariate which can be either fixed or random. The conditional tail index then depends on the covariate and we proposed several estimators with their asymptotic properties. Their behavior have been approved by simulations.In the second part, we were interested in multivariate extremes and more particularly in measuring the dependence between them. In a case of near asymptotic independence, we have to introduce new models in order to measure the dependence properly. In this context, we adapted a geostatistical tool, the madogram, and studied its asymptotic properties. We completed the study with simulations and real data of precipitations.

A maioria dos modelos matemáticos desenvolvidos para eventos raros são baseados em modelos probabilísticos para extremos. Embora as ferramentas para modelagem estatística de extremos univariados e multivariados estejam bem desenvolvidas, a extensão dessas ferramentas para modelar extremos espaciais integra uma área de pesquisa em desenvolvimento muito ativa atualmente. A modelagem de máximos sob o domínio espacial, aplicados a dados meteorológicos é importante para a gestão adequada de riscos e catástrofes ambientais nos países que tem a sua economia profundamente dependente do agronegócio. Uma abordagem natural para tal modelagem é a teoria de extremos espaciais e o processo max-stable, caracterizando-se pela extensão de dimensões infinitas da teoria de valores extremos multivariados, podendo-se então incorporar as funções de correlação existentes na geoestatística e consequentemente, verificar a dependência extrema por meio do coeficiente extremo e o madograma. Neste trabalho descreve-se a aplicação de tais processos na modelagem da dependência de máximos espaciais de precipitação máxima mensal do estado do Paraná, com base em séries históricas observadas em estações meteorológicas. Os modelos propostos consideram o espaço euclidiano e uma transformação denominada espaço climático, que permite explicar a presença de efeitos direcionais, resultantes de padrões meteorológicos sinóticos. Essa metodologia baseia-se no teorema proposto por De Haan (1984) e nos modelos de Smith (1990) e de Schlather (2002), verifica-se também o comportamento isotrópico e anisotrópico desses modelos via simulação Monte Carlo. Estimativas são realizadas através da máxima verossimilhança pareada e os modelos são comparados usando-se o Critério de Informação Takeuchi. O algoritmo utilizado no ajuste é bastante rápido e robusto, permitindo-se uma boa eficiência computacional e estatística na modelagem da precipitação máxima mensal, possibilitando-se a modelagem dos efeitos direcionais resultantes de fenômenos ambientais. / The most mathematical models developed for rare events are based on probabilistic models for extremes. Although the tools for statistical modeling of univariate and multivariate extremes are well-developed, the extension of these tools to model spatial extremes data is currently a very active area of research. Modeling of maximum values under the spatial domain, applied to meteorological data is important for the proper management of risks and environmental disasters in the countries where the agricultural sector has great influence on the economy. A natural approach for such modeling is the theory of extreme spatial and max-stable process, characterized by infinite dimensional extension of multivariate extreme value theory, and we can then incorporate the current correlation functions in geostatistics and thus, check the extreme dependence through the extreme coefficient and the madogram. This thesis describes the application of such procedures in the modeling of spatial maximum dependency of monthly maximum rainfall of Paraná State, historical series based on observed meteorological stations. The proposed models consider the Euclidean space and a transformation called climatic space, which makes it possible to explain the presence of directional effects resulting from synoptic weather patterns. This methodology is based on the theorem proposed by De Haan (1984) and Smith (1990) models and Schlather (2002), checking the isotropic and anisotropic behavior these models through Monte Carlo simulation. Estimates are performed using maximum pairwise likelihood and the models are compared using the Takeuchi information criterion. The algorithm used in the fit is very fast and robust, allowing a good statistical and computational efficiency in monthly maximum rainfall modeling, allowing the modeling of directional effects resulting from environmental phenomena.

A maioria dos modelos matemáticos desenvolvidos para eventos raros são baseados em modelos probabilísticos para extremos. Embora as ferramentas para modelagem estatística de extremos univariados e multivariados estejam bem desenvolvidas, a extensão dessas ferramentas para modelar extremos espaciais integra uma área de pesquisa em desenvolvimento muito ativa atualmente. A modelagem de máximos sob o domínio espacial, aplicados a dados meteorológicos é importante para a gestão adequada de riscos e catástrofes ambientais nos países que tem a sua economia profundamente dependente do agronegócio. Uma abordagem natural para tal modelagem é a teoria de extremos espaciais e o processo max-stable, caracterizando-se pela extensão de dimensões infinitas da teoria de valores extremos multivariados, podendo-se então incorporar as funções de correlação existentes na geoestatística e consequentemente, verificar a dependência extrema por meio do coeficiente extremo e o madograma. Neste trabalho descreve-se a aplicação de tais processos na modelagem da dependência de máximos espaciais de precipitação máxima mensal do estado do Paraná, com base em séries históricas observadas em estações meteorológicas. Os modelos propostos consideram o espaço euclidiano e uma transformação denominada espaço climático, que permite explicar a presença de efeitos direcionais, resultantes de padrões meteorológicos sinóticos. Essa metodologia baseia-se no teorema proposto por De Haan (1984) e nos modelos de Smith (1990) e de Schlather (2002), verifica-se também o comportamento isotrópico e anisotrópico desses modelos via simulação Monte Carlo. Estimativas são realizadas através da máxima verossimilhança pareada e os modelos são comparados usando-se o Critério de Informação Takeuchi. O algoritmo utilizado no ajuste é bastante rápido e robusto, permitindo-se uma boa eficiência computacional e estatística na modelagem da precipitação máxima mensal, possibilitando-se a modelagem dos efeitos direcionais resultantes de fenômenos ambientais. / The most mathematical models developed for rare events are based on probabilistic models for extremes. Although the tools for statistical modeling of univariate and multivariate extremes are well-developed, the extension of these tools to model spatial extremes data is currently a very active area of research. Modeling of maximum values under the spatial domain, applied to meteorological data is important for the proper management of risks and environmental disasters in the countries where the agricultural sector has great influence on the economy. A natural approach for such modeling is the theory of extreme spatial and max-stable process, characterized by infinite dimensional extension of multivariate extreme value theory, and we can then incorporate the current correlation functions in geostatistics and thus, check the extreme dependence through the extreme coefficient and the madogram. This thesis describes the application of such procedures in the modeling of spatial maximum dependency of monthly maximum rainfall of Paraná State, historical series based on observed meteorological stations. The proposed models consider the Euclidean space and a transformation called climatic space, which makes it possible to explain the presence of directional effects resulting from synoptic weather patterns. This methodology is based on the theorem proposed by De Haan (1984) and Smith (1990) models and Schlather (2002), checking the isotropic and anisotropic behavior these models through Monte Carlo simulation. Estimates are performed using maximum pairwise likelihood and the models are compared using the Takeuchi information criterion. The algorithm used in the fit is very fast and robust, allowing a good statistical and computational efficiency in monthly maximum rainfall modeling, allowing the modeling of directional effects resulting from environmental phenomena.

Les catastrophes naturelles comme les canicules, les tempêtes ou les précipitations extrêmes, proviennent de processus physiques et ont, par nature, une dimension spatiale ou spatiotemporelle. Le développement de modèles et de méthodes d'inférences pour ces processus est un domaine de recherche très actif. Cette thèse traite de l'inférence statistique pour les événements extrêmes dans le cadre spatial et spatio-temporel. En particulier, nous nous intéressons à deux classes de processus stochastique: les processus spatiaux max-mélange et les processus max-stable spatio-temporels. Nous illustrons les résultats obtenus sur des données de précipitations dans l'Est de l'Australie et dans une région de la Floride aux Etats-Unis. Dans la partie spatiale, nous proposons deux tests sur le paramètre de mélange a d'un processus spatial max-mélange: le test statistique Za et le rapport de vraisemblance par paire LRa. Nous comparons les performances de ces tests sur simulations. Nous utilisons la vraisemblance par paire pour l'estimation. Dans l'ensemble, les performances des deux tests sont satisfaisantes. Toutefois, les tests rencontrent des difficultés lorsque le paramètre a se situe à la frontière de l'espace des paramètres, i.e., a ∈ {0,1}, dues à la présence de paramètre de “nuisance” qui ne sont pas identifiés sous l'hypothèse nulle. Nous appliquons ces tests dans le cadre d'une analyse d'excès au delà d'un grand seuil pour des données de précipitations dans l'Est de l'Australie. Nous proposons aussi une nouvelle procédure d'estimation pour ajuster des processus spatiaux max-mélanges lorsqu'on ne connait pas la classe de dépendance extrêmal. La nouveauté de cette procédure est qu'elle permet de faire de l'inférence sans spécifier au préalable la famille de distributions, laissant ainsi parle les données et guider l'estimation. En particulier, la procédure d'estimation utilise un ajustement par la méthode des moindres carrés sur l'expression du Fλ-madogramme d'un modèle max-mélange qui contient les paramètres d'intérêt. Nous montrons la convergence de l'estimateur du paramètre de mélange a. Une indication sur la normalité asymptotique est donnée numériquement. Une étude sur simulation montrent que la méthode proposée améliore les coefficients empiriques pour la classe de modèles max-mélange. Nous implémentons notre procédure d'estimations sur des données de maximas mensuels de précipitations en Australie dans un but exploratoire et confirmatoire. Dans la partie spatio-temporelle, nous proposons une méthode d'estimation semi-paramétrique pour les processus max-stables spatio-temporels en nous basant sur une expression explicite du F-madogramme spatio-temporel. Cette partie permet de faire le pont entre la géostatistique et la théorie des valeurs extrêmes. En particulier, pour des observations sur grille régulière, nous estimons le F-madogramme spatio-temporel par sa version empirique et nous appliquons une procédure basée sur les moments pour obtenir les estimations des paramètres d'intérêt. Nous illustrons les performances de cette procédure par une étude sur simulations. Ensuite, nous appliquons cette méthode pour quantifier le comportement extrêmal de maximum de données radar de précipitations dans l'Etat de Floride. Cette méthode peut être une alternative ou une première étape pour la vraisemblance composite. En effet, les estimations semi-paramétriques pourrait être utilisées comme point de départ pour les algorithmes d'optimisation utilisés dans la méthode de vraisemblance par paire, afin de réduire le temps de calcul mais aussi d'améliorer l'efficacité de la méthode / Natural hazards such as heat waves, extreme wind speeds, and heavy rainfall, arise due to physical processes and are spatial or spatio-temporal in extent. The development of models and inference methods for these processes is a very active area of research. This thesis deals with the statistical inference of extreme and rare events in both spatial and spatio-temporal settings. Specifically, our contributions are dedicated to two classes of stochastic processes: spatial max-mixture processes and space-time max-stable processes. The proposed methodologies are illustrated by applications to rainfall data collected from the East of Australia and from a region in the State of Florida, USA. In the spatial part, we consider hypothesis testing for the mixture parameter a of a spatial maxmixture model using two classical statistics: the Z-test statistic Za and the pairwise likelihood ratio statistic LRa. We compare their performance through an extensive simulation study. The pairwise likelihood is employed for estimation purposes. Overall, the performance of the two statistics is satisfactory. Nevertheless, hypothesis testing presents some difficulties when a lies on the boundary of the parameter space, i.e., a ∈ {0,1}, due to the presence of additional nuisance parameters which are not identified under the null hypotheses. We apply this testing framework in an analysis of exceedances over a large threshold of daily rainfall data from the East of Australia. We also propose a novel estimation procedure to fit spatial max-mixture processes with unknown extremal dependence class. The novelty of this procedure is to provide a way to make inference without specifying the distribution family prior to fitting the data. Hence, letting the data speak for themselves. In particular, the estimation procedure uses nonlinear least squares fit based on a closed form expression of the so-called Fλ-madogram of max-mixture models which contains the parameters of interest. We establish the consistency of the estimator of the mixing parameter a. An indication for asymptotic normality is given numerically. A simulation study shows that the proposed procedure improves empirical coefficients for the class of max-mixture models. In an analysis of monthly maxima of Australian daily rainfall data, we implement the proposed estimation procedure for diagnostic and confirmatory purposes. In the spatio-temporal part, based on a closed form expression of the spatio-temporal Fmadogram, we suggest a semi-parametric estimation methodology for space-time max-stable processes. This part provides a bridge between geostatistics and extreme value theory. In particular, for regular grid observations, the spatio-temporal F-madogram is estimated nonparametrically by its empirical version and a moment-based procedure is applied to obtain parameter estimates. The performance of the method is investigated through an extensive simulation study. Afterward, we apply this method to quantify the extremal behavior of radar daily rainfall maxima data from a region in the State of Florida. This approach could serve as an alternative or a prerequisite to pairwise likelihood estimation. Indeed, the semi-parametric estimates could be used as starting values for the optimization algorithm used to maximize the pairwise log-likelihood function in order to reduce the computational burden and also to improve the statistical efficiency