New estimators for the extremal index and other cluster characteristics
type: Item
instance of: scholarly article
title: New estimators for the extremal index and other cluster characteristics
zbMATH Open document ID: 1053.62065
DOI: 10.1023/B:EXTR.0000031179.49454.90
author: Fabrizio Laurini
author: Jonathan A. Tawn
published in: Extremes
publication date: 2004-09-24T00:00:00Z
review text: Let \(\{X_n \}\) be a stationary sequence of rv's satisfying the asymptotic independence condition \(D(u_n)\) of \textit{M.R. Leadbetter, G. Lindgren} and \textit{H. Rootzen} [Extremes and related properties of random sequences and processes. (1983; Zbl 0518.60021)]. Let \(\{X^*_n \}\) be the associated sequence of independent rv's. As known, \[ P( max_{1\leq k\leq n}X_k \leq u_n ) \rightarrow G^{\theta}(x) \Longleftrightarrow P( max_{1\leq k\leq n} X^*_k \leq u_n ) \rightarrow G(x), \] as \(n\rightarrow \infty\). Thus, the extremal index \(0 \leq \theta \leq 1\) may be seen as a measure of dependence in extreme values. Estimating \(\theta\) is equivalent to identifying independent clusters. Different understanding of clusters leads to different estimators for \(\theta\). The run estimator of \(\theta\), \(\theta(u,m) = P( max_{2\leq k\leq m}X_k \leq u \mid X_1 >u)\), based on the characterization of \textit{G.L. O'Brien} [see Ann. Probab. 15 281--291 (1987; Zbl 0619.60025)], takes independent clusters to be exceedances of \(u\) that are separated by \(m-1\) consecutive values which do not exceed \(u\). The authors' two-thresholds extension \(\theta(u,m,c), \quad c<u \), of the runs approach has clusters terminating if a sufficiently small (i.e., \( \leq c \) ) value of the process is observed between two exceedances of the threshold \(u\). The authors consider two types of dependence: induced by either autoregression or by volatility. On different examples they illustrate the advantages of their new method for identifying clusters. Furthermore, the authors apply the method to real data for the Wooster daily minimum temperatures and the FTSE100 daily returns series.
reviewed by: Elisaveta Pancheva
Mathematics Subject Classification ID: 62G32
Mathematics Subject Classification ID: 60G70
zbMATH DE Number: 2102873
zbMATH Keywords: clusters
zbMATH Keywords: extreme value theory
zbMATH Keywords: extremal index
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2102873