A general class of estimators of the extreme value index
type: Item
instance of: scholarly article
title: A general class of estimators of the extreme value index
zbMATH Open document ID: 0929.62034
DOI: 10.1016/S0378-3758(97)00076-1
author: Holger Drees
published in: Journal of Statistical Planning and Inference
publication date: 2000-01-30T00:00:00Z
review text: A sequence of i.i.d. variables having common distribution \(F\) is observed. This distribution belongs to the weak domain of attraction of some extreme value distribution \(G\). That means \(\displaystyle{\mathcal L} (a_n^{-1}(\max_{1\leq i\leq n} X_i-b_n))\rightarrow G\) for some sequences \(a_n\) and \(b_n\). It is well known that \(G\) has the form \[ G_{\beta}(x)=exp(-(1+\beta x)^{1/\beta}),\quad 1+\beta x>0,\quad \beta\in {\mathbb R}. \] The paper under review studies different estimators of the so-called extreme value index \(\beta\). Naturally, the estimators must use only large observations. Hence the author considers estimators of the form \(\hat{\beta}=T(Q_n)\), where \(T\) is a functional and \(Q_n\) is the empirical tail quantile function. Almost all the usual estimators in the literature of the extreme value index are of this type. The method for studying the asymptotic behavior of such estimators is based on a theorem which deals with the Gaussian approximation of the empirical tail quantile function. The results have different proofs if \(\beta>0\) or \(\beta\leq 0\). In section 3, the case \(\beta>0\) is considered obtaining asymptotic normality under several conditions for the distribution function \(F\), and also a differentiability hypothesis for the functional \(T\). In the case \(\beta \leq 0\) the author needs to impose a condition on the local behavior of the functional \(T\) in a neighborhood of the constant function \(1\) to obtain asymptotic normality.
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 1115612
zbMATH Keywords: extreme value distributions
zbMATH Keywords: extreme value index
zbMATH Keywords: statistical tail functionals
reviewed by: José R. León
MaRDI profile type: Publication
cites work: Kernel estimates of the tail index of a distribution
cites work: Tail estimates motivated by extreme value theory
cites work: A moment estimator for the index of an extreme-value distribution
cites work: On the estimation of the extreme-value index and large quantile estimation
cites work: Optimal choice of sample fraction in extreme-value estimation
cites work: Refined Pickands estimators of the extreme value index
cites work: Refined pickands estimators wtth bias correction
cites work: Limit theorems for tail processes with application to intermediate quantile estimation
cites work: Von Mises calculus for statistical functionals
cites work: Q4162318
cites work: Sur la distribution limite du terme maximum d'une série aléatoire
cites work: Q3681644
cites work: On the estimation of high quantiles
cites work: Q4342744
cites work: On asymptotic normality of Hill's estimator for the exponent of regular variation
cites work: Q3670359
cites work: Best attainable rates of convergence for estimates of parameters of regular variation
cites work: Adaptive estimates of parameters of regular variation
cites work: A simple general approach to inference about the tail of a distribution
cites work: Statistical inference using extreme order statistics
cites work: Approximate distributions of order statistics. With applications to nonparametric statistics
cites work: Estimating tails of probability distributions
description: scientific article; zbMATH DE number 1115612