Characterization of valid auxiliary functions for representations of extreme value distributions and their max-domains of attraction

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.