  A geometric representation for multivariate extremes, based on the shapes of
scaled sample clouds in light-tailed margins and their so-called limit sets,
has recently been shown to connect several existing extremal dependence
concepts. However, these results are purely probabilistic, and the geometric
approach itself has not been fully exploited for statistical inference. We
outline a method for parametric estimation of the limit set shape, which
includes a useful non/semi-parametric estimate as a pre-processing step. More
fundamentally, our approach provides a new class of asymptotically-motivated
statistical models for the tails of multivariate distributions, and such models
can accommodate any combination of simultaneous or non-simultaneous extremes
through appropriate parametric forms for the limit set shape. Extrapolation
further into the tail of the distribution is possible via simulation from the
fitted model. A simulation study confirms that our methodology is very
competitive with existing approaches, and can successfully allow estimation of
small probabilities in regions where other methods struggle. We apply the
methodology to two environmental datasets, with diagnostics demonstrating a
good fit.
