We present a new framework for modelling multivariate extremes, based on an angular-radial representation of the probability density function. Under this representation, the problem of modelling multivariate extremes is transformed to that of modelling an angular density and the tail of the radial variable, conditional on angle. Motivated by univariate theory, we assume that the tail of the conditional radial distribution converges to a generalised Pareto (GP) distribution. To simplify inference, we also assume that the angular density is continuous and finite and the GP parameter functions are continuous with angle. We refer to the resulting model as the semi-parametric angular-radial (SPAR) model for multivariate extremes. We consider the effect of the choice of polar coordinate system and introduce generalised concepts of angular-radial coordinate systems and generalised scalar angles in two dimensions. We show that under certain conditions, the choice of polar coordinate system does not affect the validity of the SPAR assumptions. However, some choices of coordinate system lead to simpler representations. In contrast, we show that the choice of margin does affect whether the model assumptions are satisfied. In particular, the use of Laplace margins results in a form of the density function for which the SPAR assumptions are satisfied for many common families of copula, with various dependence classes. We show that the SPAR model provides a more versatile framework for characterising multivariate extremes than provided by existing approaches, and that several commonly-used approaches are special cases of the SPAR model. Moreover, the SPAR framework provides a means of characterising all `extreme regions' of a joint distribution using a single inference. Applications in which this is useful are discussed.

翻译：我们提出了一种基于概率密度函数角度-径向表示的多元极值建模新框架。在该表示下，多元极值建模问题被转化为对角度密度的建模以及对径向变量尾部（基于角度条件）的建模。受单变量理论启发，我们假定条件径向分布的尾部收敛于广义帕累托（GP）分布。为简化推断，我们还假定角度密度连续且有界，且GP参数函数随角度连续变化。我们将所得模型称为多元极值的半参数角度-径向（SPAR）模型。我们考察了极坐标系选择的影响，并引入了二维空间中角度-径向坐标系与广义标量角度的推广概念。研究表明，在特定条件下，极坐标系的选择不影响SPAR假设的有效性，但某些坐标系选择能产生更简洁的表示。相比之下，边际分布的选择会影响模型假设是否成立：采用拉普拉斯边际分布时，SPAR假设对多种常见依赖类别的连接函数族均成立。我们证明SPAR模型比现有方法能提供更灵活的多元极值特征刻画框架，且若干常用方法均为SPAR模型的特殊情形。此外，SPAR框架可通过单次推断刻画联合分布的所有"极值区域"。文中讨论了该模型具有实用价值的应用场景。