In this paper we study two important representations for extreme value distributions and their max-domains of attraction (MDA), namely von Mises representation (vMR) and variation representation (VR), which are convenient ways to gain limit results. Both VR and vMR are defined via so-called auxiliary functions $\psi$. Up to now, however, the set of valid auxiliary functions for vMR has neither been characterized completely nor separated from those for VR. We contribute to the current literature by introducing “universal” auxiliary functions which are valid for both VR and vMR representations for the entire MDA distribution families. Then we identify exactly the sets of valid auxiliary functions for both VR and vMR. Moreover, we propose a method for finding appropriate auxiliary functions with analytically simple structure and provide them for several important distributions.

中文翻译：

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用于表示极值分布及其最大吸引力域的有效辅助函数的表征

在本文中，我们研究了极值分布及其最大吸引域（MDA）的两种重要表示，即冯·米塞斯表示（vMR）和变分表示（VR），它们是获得极限结果的便捷方法。VR和vMR都是通过所谓的辅助函数定义的$\psi$。然而，到目前为止，vMR的有效辅助功能集既没有被完全表征，也没有与VR的辅助功能分开。我们通过引入“通用”辅助函数为当前文献做出贡献，这些函数对于整个 MDA 发行系列的VR和vMR表示都有效。然后我们准确地识别VR和vMR的有效辅助函数集。此外，我们提出了一种通过分析简单的结构寻找适当辅助函数的方法，并将其提供给几个重要的分布。