Almost sure convergence of the stable tail empirical dependence function in multivariate extreme statistics
type: Item
instance of: scholarly article
title: Almost sure convergence of the stable tail empirical dependence function in multivariate extreme statistics
zbMATH Open document ID: 0904.62061
DOI: 10.1007/BF02015138
author: Yongcheng Qi
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 1997-09-21T00:00:00Z
review text: Consider a bivariate i.i.d. sequence \((X_1Y_1)\), \((X_2,Y_2),\dots\) with joint d.f. \(F(x,y)\). Assume that \(F\) has a ``stable tail dependence function'' \(l(x,y)\), i.e. for \(x,y>0\), \[ l(x,y):= \lim_{t\downarrow 0} t^{-1} \{1-F(Q_1(tx), Q_2(ty))\} \] exists, where \(Q_i(x)= \sup\{y: 1-F_i(x)\geq x\}\), \(0\leq x\leq 1\) \((i=1,2)\). The author's main result proves that \[ l_n(x,y)= k^{-1} \sum_{j=1} I(X_j\geq X_{n-[kx]+1,n} \text{ or } Y_j\geq Y_{n-[ky]+1,n}), \] \(1\leq k\leq n\), \(k= k(n)\), \(k/\log\log n\to\infty\), but \(k/n\to 0\) (as \(n\to\infty\)), is a strongly consistent estimator of \(l(x,y)\), uniformly on \([0,T]\times [0,T]\) for any \(T>0\). This extends an earlier result of \textit{X. Huang} [Statistics of bivariate extreme values. Ph.D. Diss., Erasmus Univ., Rotterdam (1992)] who e.g. proved convergence in probability. A possible generalization to a higher-dimensional setting is also discussed.
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 1063728
zbMATH Keywords: multivariate extremes
zbMATH Keywords: stable tail empirical dependence function
zbMATH Keywords: almost sure convergence
zbMATH Keywords: strong consistency
reviewed by: J. G. Steinebach
MaRDI profile type: Publication
cites work: A moment estimator for the index of an extreme-value distribution
cites work: Q3670359
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full work available at URL: https://doi.org/10.1007/bf02015138
OpenAlex ID: W1993927541
description: scientific article; zbMATH DE number 1063728