Dependence modeling via copulas: inference, theoretical study and applications

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Copula theory provides powerful and flexible tools helping to model phenomena that imply many random variables. The proposed research project concerns the use of copulas in various contexts of application. It is divided in four parts: First, statistical methods that allow the selection of an appropriate family of copulas will be developed. The classical situation of independent and identically distributed observations, as well as the time series context, will be treated. The asymptotic validity of the proposed techniques will be demonstrated using the  theory of empirical processes. Special attention will be given to applications in the domain of statistical hydrology, where multivariate time-dependent observations are often collected. In the second part, tests for the independence between random variables and/or vectors will be proposed and evaluated from the point of view of their asymptotic efficiency. Appropriate versions of the empirical copula process will enable to generate interesting classes of tests of independence in these contexts. In the third part of the project, the dependence structures induce by multivariate conditional distributions will be investigated. The special cases of Archimedean and elliptical distributions will be studied in details. These laws exhibit the interesting property of remaining in the same class of models when a conditioning is applied. Finally, the fourth part of the project will be devoted to an important problem in finance and actuarial science, namely the risk evaluation of a portfolio of many assets. Bounds on the risk that balance for the lack of information on the nature of the dependence among the assets will be computed for various portfolios.