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Bridging Extremes: The Invertible Bimodal Gumbel Distribution.

Cira G Otiniano1, Eduarda B Silva1, Raul Y Matsushita1

  • 1Department of Statistics, University of Brasília, Brasília 70910-900, Brazil.

Entropy (Basel, Switzerland)
|December 23, 2023
PubMed
Summary

A new invertible bimodal Gumbel distribution models both maximum and minimum extremes. This versatile statistical tool offers a simple closed-form function, enhancing its use in finance and extreme value analysis.

Keywords:
Gumbel distributionbimodalityextreme value theoryvalue at risk

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Area of Science:

  • Statistics
  • Extreme Value Theory
  • Financial Mathematics

Background:

  • Traditional extreme value models often focus on either maxima or minima, limiting their application in scenarios requiring simultaneous analysis of both.
  • Existing bimodal Gumbel distributions lack a simple closed-form cumulative distribution function, posing computational challenges.

Purpose of the Study:

  • Introduce a novel three-parameter invertible bimodal Gumbel distribution.
  • Provide a statistically versatile tool for simultaneously modeling maximum and minimum extremes.
  • Enhance computational applicability in fields like finance, hydrology, and meteorology.

Main Methods:

  • Development of a novel three-parameter invertible bimodal Gumbel distribution.
  • Derivation of a simple closed-form cumulative distribution function.
  • Mathematical formulation and graphical illustration of distributional properties.
  • Application to financial data for Value at Risk (VaR) estimation.

Main Results:

  • The proposed distribution offers a closed-form CDF, simplifying calculations.
  • The model effectively captures simultaneous extreme behavior (both maxima and minima).
  • Successful estimation of Value at Risk (VaR) using financial data demonstrates practical utility.

Conclusions:

  • The invertible bimodal Gumbel distribution is a computationally attractive and versatile tool for extreme value analysis.
  • It addresses limitations of existing models by handling both maximum and minimum extremes simultaneously.
  • The model shows practical applicability, particularly in financial risk management.