A location invariant Hill-type estimator
type: Item
instance of: scholarly article
title: A location invariant Hill-type estimator
zbMATH Open document ID: 1053.62063
DOI: 10.1023/A:1015226104400
published in: Extremes
publication date: 2004-11-24T00:00:00Z
review text: A new estimator, under a semiparametric approach, which behaves similarly to Hill's estimator, see \textit{B.M. Hill}, Ann. Stat. 3, 1163--1174 (1975; Zbl 0323.62033), but which is invariant to location transformations, is studied. Let \(X_1,X_2,\ldots,X_ n\) be independent random variables with common distribution function \(F\) and let \(X_{(1,n)},X_{(2,n)},\ldots,X_{(n,n)}\) be the associated increasing order statistics. Suppose that \(F\) is in the domain of attraction of an extreme value distribution \(G_{\gamma}\) with \[ G_{\gamma}(z)=\begin{cases} \exp(-(1+\gamma z)^{-1/\gamma}),& \text{for } 1+\gamma z>0, \text{ if } \gamma\neq0\\ \exp(-\exp(-z)),& \text{for } z\in R, \text{ if } \gamma=0.\end{cases} \] The author proposes the following estimator. For \(\gamma>0\) and with an integer \(k_0\) (smaller than \(k\)) define \[ \widehat\gamma^{H}_{n}(k_0,k):= k_0^{-1} \sum_{i=0}^{k_0-1} \log (X_{(n-i,n)}-X_{(n-k,n)})/ (X_{(n-k_0,n)}-X_{(n-k,n)}). \] If \(k=k(n)\to\infty\) and \(k_0=k_0(n)\to\infty\), as \(n\to\infty\), such that \(k/n\to0\) and \(k_0/k\to0\), then \(\lim_{n\to\infty} \widehat\gamma^{H}_{n}(k_0,k)=\gamma\) in probability. It is proved that under some conditions \(\sqrt{k_0} \{\widehat\gamma^{H}_{n} (k_0,k)-\gamma\}@> d>> Z\), as \(n\to\infty\), where \(Z\) is normally distributed with mean zero and variance \(\gamma^2\). This new Hill-type estimator is location invariant. A preliminary study is carried out, with the main objective of comparing this new procedure with some other traditional estimators, namely moment and Pickands', with the consideration of locations different from zero.
Mathematics Subject Classification ID: 62G32
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 2118660
zbMATH Keywords: extreme value theory
zbMATH Keywords: tail index
zbMATH Keywords: parameter estimation
zbMATH Keywords: regular variation
author: M. Isabel Fraga Alves
reviewed by: Aleksandr D. Borisenko
MaRDI profile type: Publication
full work available at URL: a:1015226104400
OpenAlex ID: W28389390
description: scientific article; zbMATH DE number 2118660