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.

翻译：我们提出了一种基于概率密度函数的角径表示来建模多元极值的新框架。在该表示下，多元极值建模问题转化为建模角度密度以及径向变量在角度条件下的尾部问题。受一元理论启发，我们假设条件径向分布的尾部收敛于广义帕累托分布。为简化推断，我们还假设角度密度是连续有限的，且广义帕累托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径模型。我们考虑了极坐标系选择的影响，并引入了二维情形下角径坐标系和广义标量角度的广义概念。我们证明，在某些条件下，极坐标系的选择不影响SPAR假设的有效性。然而，某些坐标系的选择会导致更简单的表示。相比之下，我们证明边缘分布的选择会影响模型假设是否成立。特别地，使用拉普拉斯边缘分布得到的密度函数形式，使得SPAR假设适用于多种常见依赖类别的联结函数族。我们证明，SPAR模型为刻画多元极值提供了比现有方法更通用的框架，且几种常用方法都是SPAR模型的特例。