Baseline Methods for the Parameter Estimation of the Generalized Pareto Distribution

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.