Probabilistic and statistical aspects of multivariate
extreme value modelling

Due to the lack of a unique way
to order multivariate data, there are different ways in
which one can describe and study multivariate extremes.
Instead of following the route of classical multivariate
extreme value theory (EVT) which studies asymptotic
behaviour of coordinatewise extremes, my focus was on
the asymptotic behaviour of the shape of suitably scaled
random samples, or sample clouds. This gives a
description of the global behaviour of the data. If
sample clouds converge onto some deterministic limit set, the
boundary of this limit set will give an intuitive
picture of the behaviour of multivariate extremes across
different directions. Moreover, geometry of the limit
set can be used to determine tail dependence properties of the
underlying distribution.

Multivariate modeling beyond normality

Mathematical tractability of the multivariate normal
distribution is perhaps one of the reasons behind its
popularity in many areas of application. However,
empirical evidence often lends little support for such an
assumption. Some of the salient features of the Gaussian
distribution include light tails, elliptical symmetry and
asymptotic independence, and hence departures from
normality might be due to the lack of at least one of
these properties. I have worked with several general
classes of multivariate models, which aim to address these
shortcomings of the Gaussian distribution with a primary
focus on the last two properties.

Applications to quantitative risk management
in finance, insurance, hydrology
and geosciences