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Baseline Methods for the Parameter Estimation of the Generalized Pareto Distribution.

Jacinto Martín1, María Isabel Parra1, Mario Martínez Pizarro2

  • 1Departamento de Matemáticas, Facultad de Ciencias, Universidad de Extremadura, 06006 Badajoz, Spain.

Entropy (Basel, Switzerland)
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Summary

New Bayesian methods improve Generalized Pareto Distribution (GPD) parameter estimation by utilizing entire datasets, unlike traditional methods that waste information. These novel approaches enhance accuracy for extreme value analysis.

Keywords:
Bayesian inferenceMetropolis–Hastings algorithmextreme value theorygeneralized Pareto distributionstable distributions

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Area of Science:

  • Statistics
  • Extreme Value Theory
  • Bayesian Inference

Background:

  • Current parameter estimation methods for extreme value distributions, such as the peaks-over-threshold method for Generalized Pareto Distribution (GPD), often discard significant amounts of data by only considering observations above a specific threshold.
  • This data wastage can lead to suboptimal accuracy in parameter estimation, particularly in the context of Bayesian analysis.

Purpose of the Study:

  • To develop and present novel Bayesian methods for estimating GPD parameters that leverage the entire dataset, thereby improving estimation accuracy.
  • To incorporate information from the baseline distribution and its relationship with GPD parameters to construct highly informative priors.

Main Methods:

  • Two new Bayesian methods are proposed for GPD parameter estimation, utilizing the full baseline data and established relationships between baseline and GPD parameters to define informative priors.
  • A comparison is conducted between the proposed methods and the standard Bayesian Metropolis-Hastings algorithm using data above a threshold.
  • The study considers stable distributions (Normal, Lévy, Cauchy) as baseline distributions to explore different tail behaviors and their impact on estimation.

Main Results:

  • The proposed Bayesian methods demonstrate improved parameter estimation accuracy for GPD compared to the traditional Bayesian Metropolis-Hastings algorithm that uses only threshold-exceeding data.
  • The methods are shown to be applicable to various baseline distributions, including stable distributions and empirically fitted Gamma distributions, as evidenced by real-world air pollution data analysis.
  • The study highlights the effectiveness of utilizing the entire dataset and informative priors derived from baseline distribution properties.

Conclusions:

  • The developed Bayesian methods offer a significant improvement in the accuracy of Generalized Pareto Distribution parameter estimation by fully exploiting available data.
  • These methods provide a more efficient approach to extreme value analysis, particularly when dealing with limited extreme data points.
  • The findings suggest broader applicability of these techniques across different statistical modeling scenarios requiring robust parameter estimation.