Construction et Estimation de copules en grande dimension

In the last decades, copulas have been more and more used in statistical modeling. Their popularity owes much to the fact that they allow to separate the analysis of the margins from the analysis of the dependence structure induced by the underlying distribution. This renders easier the modeling of non Gaussian distributions, and, in particular, it allows to take into account non linear dependencies between random variables. Finance and hydrology are two examples of scientific fields where the use of copulas is nowadays standard.  Since there exists many families of bivariate copulas, it is always possible for the user to choose one that suits his needs. Unfortunately, the multivariate case is not that simple. The choice of a multivariate model always goes along with more or less severe drawbacks. Moreover, there are only a few copula multivariate models. This thesis addresses this issue. We propose two classes of multivariate copulas with novel properties, resulting in an enlargement of the range of the existing models. The first model writes as a product of bivariate copulas and is underlain by a tree structure where each edge represents a bivariate copula. Hence, we are able to model different pairs with different dependence properties. The second one is a factor model, with a singular component, built on a nonparametric class of bivariate copulas. It exhibits a good balance between tractability and flexibility. Since the copulas belonging to the second proposed class have a singular component, the standard methods of inference do not permit to estimate their parameters. For this reason – and this is a contribution of our thesis as well –, we also deal with the estimation of copulas in general, and establish the asymptotic properties of a least-squares estimator based on dependence coefficients without imposing regularity conditions on the copulas. The models and methods have been applied to hydrological data (flow rates and rain falls).

Directeurs:

• Mme Florence Forbes (Directeur de recherche - INRIA )
• Mr Stéphane Girard (Chargé de Recherche - INRIA )

Raporteurs:

• Mr Johan Segers (Professeur - Université Catholique de Louvain )
• Mr Fabrizio Durante (Assistant Professeur - Free University of Bolzen-Balzano )

Examinateurs:

• Mme Anne Favre-Pugin (Professeur - Université Joseph Fourier )
• Mr Ivan Kojadinovic (Professeur - Université de Pau et des Pays de l'Adour )