Multivariate Data-based Risk Measures

In [12], the concept of natural risk statistics is introduced as a data-based risk measure, i.e. as an axiomatic risk measure defined in the space Rn. In this note, we set to generalize this notion to bivariate data sets (more generally, multivariate data sets) by defining an axiomatic risk measure on the space RnRn. This construction requires the definition of a suitable pair-order in R2. We propose an aggregate-loss order that allows us to give a coherent characterization of these bivariate natural risk statistics. Some examples and applications of these measures are provided.