Probabilistic and statistical aspects of extremes of univariate processes have been extensively studied, and recent developments in extremes have focused on multivariate theory and its application. Multivariate extreme value theory encompasses two separate aspects: marginal features, which may be handled by standard univariate methods, and dependence features. Both will be examined in this study. First we focus on testing independence in multivariate extremes. All existing score tests of independence in multivariate extreme values have non-regular properties that arise due to violations of the usual regularity conditions of maximum likelihood. Some of these violations may be dealt with using standard techniques, for example when independence corresponds to a boundary point of the parameter space of the underlying model. However, another type of regularity violation, the infinite second moment of the score function, is more difficult to deal with and has important consequences for applications, resulting in score statistics with non-standard normalisation and poor rates of convergence. We propose a likelihood based approach that provides asymptotically normal score tests of independence with regular normalisation and rapid convergence. The resulting tests are straightforward to implement and are beneficial in practical situations with realistic amounts of data. A fundamental issue in applied multivariate extreme value (MEV) analysis is modelling dependence within joint tail regions. The primary aim of the remainder of this thesis is to develop a pseudo-polar framework for modelling extremal dependence that extends the existing classical results for multivariate extremes to encompass asymptotically independent tails. Accordingly, a constructional procedure for obtaining parametric asymptotically independent joint tail models is developed. The practical application of this framework is analysed through applications to bivariate simulated and environmental data, and joint estimation of dependence and marginal parameters via likelihood methodology is detailed. Inference under our models is examined and tests of extremal asymptotic independence and asymmetry are derived which are useful for model selection. In contrast to the classical MEV approach, which concentrates on the distribution of the normalised componentwise maxima, our framework is based on modelling joint tails and focuses directly on the tail structure of the joint survivor function. Consequently, this framework provides significant extensions of both the theoretical and applicable tools of joint tail modelling. Analogous point process theory is developed and the classical componentwise maxima result for multivariate extremes is extended to the asymptotically independent case. Finally, methods for simulating from two of our bivariate parametric models are provided.

La détection d'anomalies est tout d'abord une étape utile de pré-traitement des données pour entraîner un algorithme d'apprentissage statistique. C'est aussi une composante importante d'une grande variété d'applications concrètes, allant de la finance, de l'assurance à la biologie computationnelle en passant par la santé, les télécommunications ou les sciences environnementales. La détection d'anomalies est aussi de plus en plus utile au monde contemporain, où il est nécessaire de surveiller et de diagnostiquer un nombre croissant de systèmes autonomes. La recherche en détection d'anomalies inclut la création d'algorithmes efficaces accompagnée d'une étude théorique, mais pose aussi la question de l'évaluation de tels algorithmes, particulièrement lorsque l'on ne dispose pas de données labellisées -- comme dans une multitude de contextes industriels. En d'autres termes, l'élaboration du modèle et son étude théorique, mais aussi la sélection du modèle. Dans cette thèse, nous abordons ces deux aspects. Tout d'abord, nous introduisons un critère alternatif au critère masse-volume existant, pour mesurer les performances d'une fonction de score. Puis nous nous intéressons aux régions extrêmes, qui sont d'un intérêt particulier en détection d'anomalies, pour diminuer le taux de fausse alarme. Enfin, nous proposons deux méthodes heuristiques, l'une pour évaluer les performances d'algorithmes de détection d'anomalies en grande dimension, l'autre pour étendre l'usage des forets aléatoires à la classification à une classe. / Anomaly detection is not only a useful preprocessing step for training machine learning algorithms. It is also a crucial component of many real-world applications, from various fields like finance, insurance, telecommunication, computational biology, health or environmental sciences. Anomaly detection is also more and more relevant in the modern world, as an increasing number of autonomous systems need to be monitored and diagnosed. Important research areas in anomaly detection include the design of efficient algorithms and their theoretical study but also the evaluation of such algorithms, in particular when no labeled data is available -- as in lots of industrial setups. In other words, model design and study, and model selection. In this thesis, we focus on both of these aspects. We first propose a criterion for measuring the performance of any anomaly detection algorithm. Then we focus on extreme regions, which are of particular interest in anomaly detection, to obtain lower false alarm rates. Eventually, two heuristic methods are proposed, the first one to evaluate anomaly detection algorithms in the case of high dimensional data, the other to extend the use of random forests to the one-class setting.