The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In the common situation that the components of the vector have different distributions, the rank transformation offers a convenient and robust way of standardizing data in order to build an empirical version of the angular measure based on the most extreme observations. We provide a functional asymptotic expansion for the empirical angular measure in the bivariate case based on the theory of weak convergence in the space of bounded functions. From the expansion, not only can the known asymptotic distribution of the empirical angular measure be recovered, it also enables to find expansions and weak limits for other statistics based on the associated empirical process or its quantile version.

翻译：单位球面上的角测度通过标准化边际分布刻画了随机向量分量在极端区域的一阶依赖结构，其统计恢复是涉及远离中心观测的学习问题中的关键步骤。在向量分量具有不同分布的常见情形下，秩变换提供了一种便捷且稳健的数据标准化方法，从而能够基于最极端观测构建角测度的经验版本。本文基于有界函数空间中的弱收敛理论，给出了双变量情形下经验角测度的函数渐近展开式。该展开式不仅能够恢复已知的经验角测度渐近分布，还可为基于关联经验过程或其分位数版本的其他统计量提供展开式与弱极限。