We propose a model to flexibly estimate joint tail properties by exploiting the convergence of an appropriately scaled point cloud onto a compact limit set. Characteristics of the shape of the limit set correspond to key tail dependence properties. We directly model the shape of the limit set using B\'ezier splines, which allow flexible and parsimonious specification of shapes in two dimensions. We then fit the B\'ezier splines to data in pseudo-polar coordinates using Markov chain Monte Carlo, utilizing a limiting approximation to the conditional likelihood of the radii given angles. By imposing appropriate constraints on the parameters of the B\'ezier splines, we guarantee that each posterior sample is a valid limit set boundary, allowing direct posterior analysis of any quantity derived from the shape of the curve. Furthermore, we obtain interpretable inference on the asymptotic dependence class by using mixture priors with point masses on the corner of the unit box. Finally, we apply our model to bivariate datasets of extremes of variables related to fire risk and air pollution.

翻译：我们提出了一种模型，通过利用适当缩放的点云收敛到紧致极限集，灵活估计联合尾部性质。极限集形状的特征对应关键的尾部依赖性质。我们直接使用Bézier样条对极限集形状建模，该样条允许在二维空间中灵活且简约地指定形状。随后，我们利用马尔可夫链蒙特卡洛方法，基于给定角度半径的条件似然的极限近似，将Bézier样条拟合到伪极坐标下的数据。通过施加Bézier样条参数的适当约束，我们确保每个后验样本都是有效的极限集边界，从而允许对曲线形状导出的任何量进行直接后验分析。此外，通过使用在单位盒子角落处具有点质量的混合先验，我们获得了关于渐近依赖类别的可解释推断。最后，我们将模型应用于与火灾风险和空气污染相关的变量极值的双变量数据集。