Risk assessment for natural hazards and financial extreme events requires the statistical analysis of extreme events, often beyond observed levels. The characterization and extrapolation of the probability of rare events rely on assumptions about the extremal dependence type and about the specific structure of statistical models. In this thesis, we develop models with flexible tail dependence structures, in order to provide a reliable estimation of tail characteristics and risk measures. From a methodological perspective, this thesis makes the following novel developments. 1) We propose new copula-based models for multivariate and spatial extremes with flexible tail dependence structures, which are parsimonious and able to bridge smoothly asymptotic dependence and asymptotic independence classes, in both the upper and the lower tails; 2) Moreover, aiming at describing more general dependence structures using graphs, we propose a novel extremal dependence measure called the partial tail-correlation coefficient (PTCC) under the framework of regular variation to learn complex extremal network structures; 3) Finally, we develop a semi-parametric neural-network-based regression model to identify spatial causal effects at all quantile levels (including low and high quantiles). Overall, we make novel contributions to creating new flexible extremal dependence models, developing and implementing novel Bayesian computation algorithms, and taking advantage of machine learning and causal inference principles for modeling extremes. Our novel methodologies are illustrated by a range of applications to financial, climatic, and health data. Specifically, we apply our bivariate copula model to the historical closing prices of five leading cryptocurrencies and estimate the extremal dependence evolution over time, and we use the PTCC to learn the extreme risk network of historical global currency exchange data. Moreover, our multivariate spatial factor copula model is applied to study the upper and lower extremal dependence structures of the daily maximum and minimum air temperature from the state of Alabama in the southeastern United States; and we also apply the PTCC in extreme river discharge network learning for the Upper Danube basin. Finally, we apply the causal spatial quantile regression model in quantifying spatial quantile treatment effects of maternal smoking on extreme low birth weight of newborns in North Carolina, United States.

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.

In recent years, there has been a surge of research interest in the analysis of time series and spatial data. While on one hand more and more sophisticated models are being developed, on the other hand the resulting theory and estimation process has become more and more involved. This dissertation addresses the development of statistical inference procedures for data exhibiting dependencies of varied form and structure. In the first work, we consider estimation of the mean squared prediction error (MSPE) of the best linear predictor of (possibly) nonlinear functions of finitely many future observations in a stationary time series. We develop a resampling methodology for estimating the MSPE when the unknown parameters in the best linear predictor are estimated. Further, we propose a bias corrected MSPE estimator based on the bootstrap and establish its second order accuracy. Finite sample properties of the method are investigated through a simulation study. The next work considers nonparametric inference on spatial data. In this work the asymptotic distribution of the Discrete Fourier Transformation (DFT) of spatial data under pure and mixed increasing domain spatial asymptotic structures are studied under both deterministic and stochastic spatial sampling designs. The deterministic design is specified by a scaled version of the integer lattice in IRd while the data-sites under the stochastic spatial design are generated by a sequence of independent random vectors, with a possibly nonuniform density. A detailed account of the asymptotic joint distribution of the DFTs of the spatial data is given which, among other things, highlights the effects of the geometry of the sampling region and the spatial sampling density on the limit distribution. Further, it is shown that in both deterministic and stochastic design cases, for "asymptotically distant" frequencies, the DFTs are asymptotically independent, but this property may be destroyed if the frequencies are "asymptotically close". Some important implications of the main results are also given.