Séminaire MODAL'X : Gauthier Thurin (IMB, Bordeaux)

Résumé:
Every attempt to define quantiles for multivariate data face the same issue : the lack of a canonical ordering. A recent concept is based on the theory of optimal transportation, with promising geometric features and numerous applications, from statistical testing to quantile regression. On the real line, the superquantile function has received special attention, as well as the expected shortfall function, its lower-tail counterpart. These are averages of quantiles beyond or ahead a certain level, which, in the multivariate setting, does not adapt canonically.  
After an introduction to quantiles based on optimal transport, we will extend these definitions to the multivariate setting. These new functions naturally describe multivariate tail probabilities and central areas of point clouds, respectively. We will show that they characterize random vectors and their convergence in distribution. Finally, these definitions will be applied to risk measurement, with multivariate definitions of Value-at-Risk and Conditional-Value-at-Risk.