Bregman superquantiles. Estimation methods and applications

Abstract

In this work, we extend some parameters built on a probability distribution introduced before to the casewhere the proximity between real numbers is measured by using a Bregman divergence. This leads to the definition of the Bregman superquantile (thatwe can connect with severalworks in economy, see for example [18] or [9]). Axioms of a coherent measure of risk discussed previously (see [31] or [3]) are studied in the case of Bregman superquantile. Furthermore, we deal with asymptotic properties of aMonte Carlo estimator of the Bregman superquantile. Several numerical tests confirm the theoretical results and an application illustrates the potential interests of the Bregman superquantile.