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.

翻译：双变量极值事件的统计分析关键在于灵活刻画两个变量间的尾部依赖关系。极值理论文献中，基于不同渐近极限已有多种技术可分别建模尾部依赖的各个侧面。Balkema与Nolde（2010）及Nolde（2014）的研究表明，在表征联合尾部整体特征时，研究适当标度样本云的极限形状具有重要价值。本文首次发展出针对该极限集的统计推断方法，这对构建联合尾部不同侧面的统一推断框架具有重要实践意义。此外，Nolde与Wadsworth（2022）将该极限集与多种现有极值依赖框架建立联系。因此，作为极限集推断新方法的衍生成果，我们首次获得若干极值依赖测度的自洽估计量，从而规避当前可能存在的矛盾性结论。模拟实验中，我们的极限集估计器在多种分布类型下表现优异，相应的极值依赖估计量相较于现有技术实现整体显著提升与局部渐进改进。在海洋波高应用案例中，该估计方法成功捕捉到随地点间距增大而减弱的极值依赖特征。