A dynamical mixture model for unsupervised tail estimation without threshold selection
type: Item
instance of: scholarly article
title: A dynamical mixture model for unsupervised tail estimation without threshold selection
zbMATH Open document ID: 1039.62042
DOI: 10.1023/A:1024072610684
author: Arnoldo Frigessi
author: Ola Haug
author: Håvard Rue
published in: Extremes
publication date: 2004-03-16T00:00:00Z
review text: This paper deals with an unsupervised approach to tail estimation. The authors model all data, not only those belonging to the tail. They use a mixture model, one component of which is a generalized Pareto distribution and the other component has a density distribution with light tail, and the weight in the mixture is dynamic and determines when the generalized Pareto distribution component in the mixture is prevalent. Let \(X_{1},\ldots,X_{n}\) be nonnegative i.i.d. random variables with common probability density function \[ l(x)= \{[1- p(x;\theta)] f(x;\beta)+ p(x;\theta)g(x;\xi,\sigma)\}/ Z(\theta,\beta,\xi,\sigma), \] where \(g(x;\xi,\sigma)=\sigma^{-1} (1+\xi x/\sigma)^{-1/\xi-1}\) is the generalized Pareto distribution density, \(p(x;\theta)\) is the mixing function taking values in \((0,1]\), increasing in \(x\) and such that \(\lim_{x\to x_{\infty}}p(x;\theta)=1\) for all \(\theta\), \(f(x;\beta)\) is a density, and \(Z(\theta,\beta,\xi,\sigma)\) is the normalizing constant. The authors show how to choose \(f\) and \(p\) and present an algorithm for sampling from \(l\). Likelihood based inference for the model is discussed and a numerical procedure is described. The authors compare the proposed dynamic mixture methods with Dupuis' robust threshold approach in peak-over-threshold inference. Simulation results are presented that indicate that the new approach can be useful in unsupervised tail estimation, especially in heavy tailed situations and for small percentiles.
reviewed by: Aleksandr D. Borisenko
Mathematics Subject Classification ID: 62G32
Mathematics Subject Classification ID: 62F10
Mathematics Subject Classification ID: 62P05
zbMATH DE Number: 2059067
zbMATH Keywords: peaks over threshold
zbMATH Keywords: shape parameter
zbMATH Keywords: heavy tailed distribution
zbMATH Keywords: maximum likelihood
zbMATH Keywords: mixture model
MaRDI profile type: Publication
full work available at URL: a:1024072610684
OpenAlex ID: W46851279
description: scientific article; zbMATH DE number 2059067