  The key to successful statistical analysis of bivariate extreme events lies
in flexible modelling of the tail dependence relationship between the two
variables. In the extreme value theory literature, various techniques are
available to model separate aspects of tail dependence, based on different
asymptotic limits. Results from Balkema and Nolde (2010) and Nolde (2014)
highlight the importance of studying the limiting shape of an
appropriately-scaled sample cloud when characterising the whole joint tail. We
now develop the first statistical inference for this limit set, which has
considerable practical importance for a unified inference framework across
different aspects of the joint tail. Moreover, Nolde and Wadsworth (2022) link
this limit set to various existing extremal dependence frameworks. Hence, a
by-product of our new limit set inference is the first set of self-consistent
estimators for several extremal dependence measures, avoiding the current
possibility of contradictory conclusions. In simulations, our limit set
estimator is successful across a range of distributions, and the corresponding
extremal dependence estimators provide a major joint improvement and small
marginal improvements over existing techniques. We consider an application to
sea wave heights, where our estimates successfully capture the expected
weakening extremal dependence as the distance between locations increases.
