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Baseline Methods for the Parameter Estimation of the Generalized Pareto Distribution
Authors:Jacinto Martín  María Isabel Parra  Mario Martínez Pizarro  Eva Lpez Sanjun
Institution:1.Departamento de Matemáticas, Facultad de Ciencias, Universidad de Extremadura, 06006 Badajoz, Spain; (J.M.); (M.I.P.);2.Departamento de Matemáticas, Centro Universitario de Mérida, Universidad de Extremadura, 06800 Mérida, Spain;
Abstract:In the parameter estimation of limit extreme value distributions, most employed methods only use some of the available data. Using the peaks-over-threshold method for Generalized Pareto Distribution (GPD), only the observations above a certain threshold are considered; therefore, a big amount of information is wasted. The aim of this work is to make the most of the information provided by the observations in order to improve the accuracy of Bayesian parameter estimation. We present two new Bayesian methods to estimate the parameters of the GPD, taking into account the whole data set from the baseline distribution and the existing relations between the baseline and the limit GPD parameters in order to define highly informative priors. We make a comparison between the Bayesian Metropolis–Hastings algorithm with data over the threshold and the new methods when the baseline distribution is a stable distribution, whose properties assure we can reduce the problem to study standard distributions and also allow us to propose new estimators for the parameters of the tail distribution. Specifically, three cases of stable distributions were considered: Normal, Lévy and Cauchy distributions, as main examples of the different behaviors of the tails of a distribution. Nevertheless, the methods would be applicable to many other baseline distributions through finding relations between baseline and GPD parameters via studies of simulations. To illustrate this situation, we study the application of the methods with real data of air pollution in Badajoz (Spain), whose baseline distribution fits a Gamma, and show that the baseline methods improve estimates compared to the Bayesian Metropolis–Hastings algorithm.
Keywords:Bayesian inference  generalized Pareto distribution  Metropolis–  Hastings algorithm  stable distributions  extreme value theory
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