Grant: Discovery Grants Program - Individual

A thorough understanding of many financial and climatic phenomena requires the simultaneous observation of many random variables. Recently, the theory of copulas has revolutionized the classical approach of multidimensional modelling by perfectly defining the nature of the dependence between random variables without relying on the individual stochastic behaviours. However, a satisfactory and rigorous application of copulas requires the use of statistical methods that are strongly justified, which remains to be developed in many contexts of applications. The main aim of this proposal is to elaborate new statistical methodologies in order to verify hypotheses that can be stated in terms of copulas. A special attention will be given to so-called ""composite"" hypotheses. In order for the procedures to be completely rigorous, modern methods of re-sampling will be adapted for each treated situations and will be formally justified. Many important aspects will be visited, in particular tests of independence between random vectors, tests for the detection of break-points in dependence structures, nonparametric symmetry tests and goodness-of-fit procedures. The possibility to verify formally these hypotheses will enable a better understanding of many natural and financial phenomena, where these questions are of a great interest. Also, in addition to the development of new statistical procedures for specific problems, this project will allow a better use of empirical processes based on ranks. In particular, the proofs of the validity of the re-sampling techniques that will be defined, like the bootstrap and the multiplier method, will allow to consider many other interesting hypotheses in the future.