The regular variation model for multivariate extremes decomposes the joint distribution of the extremes in polar coordinates in terms of the angles and the norm of the random vector as the product of two independent densities: the angular (spectral) measure and the density of the norm. The support of the angular measure is the surface of a unit hypersphere and the density of the norm corresponds to a Pareto density. The dependence structure is determined by the angular measure on the hypersphere, and directions with high probability characterize the dependence structure among the elements of the random vector of extreme values. Previous applications of the regular variation model have not considered a probabilistic model for the angular density and no statistical tests were applied. In this paper, circular and spherical distributions based on nonnegative trigonometric sums are considered flexible probabilistic models for the spectral measure that allows the application of statistical tests to make inferences about the dependence structure among extreme values. The proposed methodology is applied to real datasets from finance.

翻译：多变量极值的正则变化模型将极值联合分布在极坐标下分解为角度与随机向量范数的乘积形式，即两个独立密度的乘积：角（谱）测度与范数密度。角测度的支撑集为单位超球面表面，范数密度对应帕累托密度。依赖结构由超球面上的角测度决定，高概率方向表征极值随机向量各分量间的依赖关系。此前正则变化模型的应用并未考虑角密度的概率模型，也未进行统计检验。本文采用基于非负三角和函数的圆分布与球分布作为谱测度的灵活概率模型，从而能够应用统计检验推断极值间的依赖结构。所提出的方法已应用于金融领域的真实数据集。