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，刻画了随机向量各分量在极端区域的依存结构的一阶性质。在涉及远离中心观测值的学习问题中，角测度的统计推断是重要环节。当向量分量具有不同分布时，秩变换提供了一种便捷稳健的标准化方法，可基于极端观测值构建角测度的经验版本。本文基于有界函数空间弱收敛理论，给出了双变量情形下经验角测度的泛函渐近展开。该展开不仅能恢复经验角测度的已知渐近分布，还可推导基于相关经验过程或其分位数版本的其他统计量的渐近展开与弱极限。