Summary: 4 Multivariate extremes
In this section we consider the problems we face if we wish to model the extremal behaviour
of two or more (dependent) processes simultaneously. There are several reasons why we may
wish to do this:
· to model the extreme behaviour of a particular variable over several nearby locations (e.g.
rainfall over a network of sites);
· to model the joint extremes of two or more different variables at a particular location (e.g.
wind and rain at a site);
· to model the joint behaviour of extremes which occur as consecutive observations in a
timeseries (e.g. consecutive hourly maximum wind gusts during a storm).
All of these problems suggest fitting an appropriate limiting multivariate distribution to the
relevant data. However, as we shall see, the derivation of such a multivariate distribution is
not as easy as we might hope. The analogy with the Normal distribution as a model for means
breaks down as we move into n dimensions! It is not even clear what the `relevant data' should
be! Most of the increased complexity is apparent in the move from 1 to 2 dimensions, so we
will focus largely on bivariate problems.
4.2 Componentwise maxima models
4.2.1 Example: network of rainfall measurements
Suppose we want to study the joint extremes of daily rainfall accumulations at the network of 8