Modeling the dependency between pre-extremes.

Summary The extreme joint behavior between random variables is of particular interest in many applications in environmental sciences, finance, insurance or risk management. For example, this behavior plays a central role in the evaluation of natural disaster risks. A misspecification of the dependence between random variables can lead to a dangerous underestimation of the risk, especially at the extreme level. The first objective of this thesis is to develop inference techniques for Archimax copulas. These dependence models can capture any type of asymptotic dependence between the extremes and, simultaneously, model the risks attached to the mean level. An Archimax copula is characterized by its two functional parameters, the stable caudal dependence function and the Archimedean generator that acts as a distortion affecting the extreme dependence regime. Conditions are derived so that the generator and the caudal function are identifiable, so that a semi-parametric inference approach can be developed. Two nonparametric estimators of the caudal function and a moment-based estimator of the generator, assuming that the latter belongs to a parametric family, are advanced. The asymptotic behavior of these estimators is then established under non-restrictive regularity assumptions and the finite sample performance is evaluated through a simulation study. A hierarchical (or "cluster") construction that generalizes the Archimax copulas is proposed in order to provide more flexibility, making it more suitable for practical applications. The extreme behavior of this new dependence model is then studied, which leads to a new way of constructing stable caudal dependence functions. The Archimax copula is then used to analyze the monthly precipitation maxima, observed at three weather stations in Brittany. The model seems to fit the data very well, both for light and heavy precipitation. The non-parametric estimator of the caudal function reveals an extreme asymmetric dependence between stations, reflecting the movement of thunderstorms in the region. An application of the hierarchical Archimax model to a precipitation dataset containing 155 stations is then presented, in which asymptotically dependent groups of stations are determined via a "clustering" algorithm specifically adapted to the model. Finally, possible methods to model inter-cluster dependence are discussed.