Optimal choice of sample fraction in extreme-value estimation
type: Item
instance of: scholarly article
title: Optimal choice of sample fraction in extreme-value estimation
zbMATH Open document ID: 0797.62016
author: Arnold L. M. Dekkers
author: Laurens De Haan
published in: Journal of Multivariate Analysis
publication date: 1994-06-06T00:00:00Z
full work available at URL: https://ir.cwi.nl/pub/5577
review text: Let \(X_ 1, \dots, X_ n\) be an iid sample from an unknown distribution \(F\). A distribution \(F\) is in the maximum domain of attraction of an extreme value distribution \(G_ \gamma\) if there exist constants \(c_ n\), \(d_ n\) such that \[ c_ n^{-1} \bigl( \max (X_ 1, \dots, X_ n)-d_ n \bigr) @>d>>G_ \gamma. \] The distribution function \(G_ \gamma(x)\) is then of the form \(\exp \{-(1 + \gamma x)^{-1/ \gamma}\}\) with the natural choice for \(x\)-values and \(\gamma \in {\mathcal R}\). The authors study in detail the asymptotic bias of moment estimators \(\widehat \gamma_ n\) for the extreme value index \(\gamma\) under natural conditions such as regular variation of the generalised inverse of \(1/(1- F)\) and its modifications and generalizations. They also consider the trade-off between bias and variance of \((\widehat \gamma_ n - \gamma)\). In particular, they determine the fraction of the upper order statistics of the sample which minimises \(\text{var} (\widehat \gamma_ n - \gamma)\).
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 60G70
zbMATH DE Number: 562369
zbMATH Keywords: Hill estimator
zbMATH Keywords: iid sample
zbMATH Keywords: maximum domain of attraction
zbMATH Keywords: extreme value distribution
zbMATH Keywords: asymptotic bias of moment estimators
zbMATH Keywords: extreme value index
zbMATH Keywords: regular variation of the generalised inverse
zbMATH Keywords: upper order statistics
reviewed by: T. Mikosch
MaRDI profile type: Publication
OpenAlex ID: W2076253795
DOI: 10.1006/JMVA.1993.1078
description: scientific article; zbMATH DE number 562369