Séminaire LJKDonnées et Aléatoire Théorie & Applications

Le Jeudi 30 Janvier 2020 à 14h00 en Salle 106  Batiment IMAG

Séminaire de Mr Gilles STUPFLER (ENSAI)

Characteristic functions as offshoots of multivariate extreme value theory

Résumé:

Multivariate extreme value theory, namely the definition and study of extreme events in several dimensions, has been largely developed over the last 30 years. One particularly interesting framework for multivariate extreme value theory is obtained by extending the univariate definition of extreme value distributions, found from the convergence of linearly normalised sample maxima, to the multivariate case via the use of the componentwise maximum operator. This gives rise to the class of multivariate maxstable distribution functions. A theory of multivariate regular variation, at the heart of which the angular measure lies, can then be written and constitutes a reasonable framework for inference about extreme events in multidimensional settings.
Although this is by no means obvious, a common thread through multivariate extreme value theory can be found using the notion of Dnorm: loosely, there is a simple, explicit onetoone correspondence between maxstable distributions with negative exponential margins and the set of Dnorms. These norms have mainly been studied for their potential to unify the presentation of multivariate extreme value theory and trivialise certain proofs, such as those of Takahashi's characterisations of complete dependence or independence of the components of a maxstable distribution. As a mathematical object themselves, however, Dnorms are interesting as well, and their theory remains in full development. I will here introduce the notion of Dnorm and then present two offshoots of multivariate extreme value theory inspired by this concept, which I will call the max and mincharacteristic function. I will present some interesting properties of such functions (they characterise distributions, are compatible with weak convergence, have simple closed forms for Generalised Pareto distributions...) and will discuss potential applications in statistics. [Joint work with Michael Falk, University of Wuerzburg]
