Abstract

In this talk, I show that the extremal dependence of $d$-dimensional random vectors can be uniquely characterized by a class of random vectors residing on a $(d-1)$-dimensional hyperplane. This translates the statistical analyses on multivariate extremes to that on a linear vector space, enabling the application of a large number of existing statistical techniques. As an example, we show that for any model, a lower-dimensional approximation can be achieved naturally through principal component analysis. Additionally, through this framework, the widely used H\"usler-Reiss family for modelling extremes is directly linked to the Gaussian family on a hyperplane, thereby justifying its status as the Gaussian counterpart for extremes.