Multivariate generalized Pareto distributions

Holger Rootzén and Nader Tajvidi

Abstract:: Statistical inference for extremes has been a subject of intensive research during the past couple of decades. One approach is based on modeling exceedances of a random variable over a high threshold with the Generalized Pareto (GP) distribution. This has shown to be an important way to apply extreme value theory in practice and is widely used. In this paper we introduce a multivariate analogue of the GP distribution and show that it is characterized by each of following two properties: (i) exceedances asymptotically has a multivariate GP distribution if and only if maxima asymptotically are EV distributed, and (ii) the multivariate GP distribution is the only one which is preserved under change of exceedance levels. We also give a number of examples and discuss lowerdimensional marginal distributions.

Key words:: Peaks over threshold method, Generalized Pareto distribution, Multivariate extreme value theory, Multivariate Pareto distribution, Non-homogeneous Poisson process