Smoothing the moment estimator of the extreme value parameter
type: Item
instance of: scholarly article
title: Smoothing the moment estimator of the extreme value parameter
zbMATH Open document ID: 0921.62028
DOI: 10.1023/A:1009925716617
author: Cătălin Stărică
author: Sidney I. Resnick
published in: Extremes
publication date: 1999-08-12T00:00:00Z
review text: Let \(\{X_n\}\) be a sequence of independent random variables whose common distribution function \(F(X)\) belongs to the domain of attraction of an extreme value distribution \(G(x)=\exp\{-(1+\gamma x)^{-1/\gamma}\}\), \(\gamma \in R^1\). This paper focused on the moment estimator \(\gamma_{k,n}\) for an extreme value parameter \(\gamma\) based on \(K\) upper order statistics, proposed by \textit{A.L.M. Dekkers, J.H.J. Einmahl} and \textit{L. de Haan} [Ann. Stat. 17, No. 4, 1833-1855 (1989; Zbl 0701.62029]. The smoothed modification \[ av\gamma_{k,n}=(1/k(1-s))\sum_{p=[ks]+1}^k \gamma_{p,n}, \] where \(s\in [0,1]\), \([x]\) entire of \(x\), is obtained through an average technique. The new estimator is based on the tail of empirical processes and this fact is utilized in investigation of its asymptotic properties. Also, \(av\gamma_{k,n}\) offers some advantages in practice; for instance, it yields a less volatile function of \(k\) and, thus, simplifies the choice of \(k\). A graphical comparison of the performance of both estimators indicates that the moment estimator is superior for \(\gamma>0\) and inferior for \(\gamma<0\). An example of application of smoothed estimation for a real data set from hydrology is presented.
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 62F10
zbMATH DE Number: 1324027
zbMATH Keywords: extreme value distributions
zbMATH Keywords: domain of attraction
zbMATH Keywords: extreme value parameters
zbMATH Keywords: moment estimators
zbMATH Keywords: smoothed estimation
zbMATH Keywords: tail of empirical processes
MaRDI profile type: Publication
full work available at URL: a:1009925716617
OpenAlex ID: W2158695583
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description: scientific article; zbMATH DE number 1324027