Paper: arXiv 2311.15355

Authors: Miriam Isabel Seifert

Abstract

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper provides a rigorous mathematical characterization of auxiliary functions in extreme value theory, filling a theoretical gap regarding max-domains of attraction. However, as a purely theoretical probability paper, it lacks empirical data, backtests, or practical implementation details.

Research Flowchart

  flowchart TD
    A[Research Goal: Characterize valid auxiliary functions psi<br/>for VR and vMR in Extreme Value Theory] --> B[Input: Distribution Families in Max-Domains<br/>of Attraction & Existing VR/vMR Frameworks]
    B --> C[Method: Mathematical Analysis of Function Classes<br/>& Representation Conditions]
    C --> D[Introduce Universal Auxiliary Functions<br/>valid for both VR and vMR]
    D --> E[Derive Analytical Method for Constructing<br/>Simple Auxiliary Functions]
    E --> F[Outcomes: Complete Sets of Valid Functions for VR/vMR<br/>& Simple Functions for Key Distributions]