Rates of convergence for bivariate extremes
type: Item
instance of: scholarly article
title: Rates of convergence for bivariate extremes
zbMATH Open document ID: 0880.62015
published in: Journal of Multivariate Analysis
publication date: 1998-02-05T00:00:00Z
review text: Let \(\{(X_n,Y_n)\), \(n\geq 1\}\) be independent, identically distributed (i.i.d.) random vectors with common distribution function \(F\). Suppose \(F\) is in the domain of attraction of an extreme value distribution, i.e., there exist constants \(a_n>0\), \(c_n>0\), \(b_n\), and \(d_n\) such that \[ F^n(a_nx+ b_n, c_ny+ d_n)\to G(x,y) \tag{1} \] for \((x,y)\in \mathbb{R}^2\), where \(G\) has non-degenerate marginal distributions, i.e., \[ P\Biggl\{ \frac{\max_{i\leq n}X_i-b_n}{a_n}\leq x, \frac{\max_{i\leq n}y_i-d_n}{c_n}\leq y\Biggr\}\to G(x,y) \qquad (n\to\infty). \] Then \(G(x,\infty)\) and \(G(\infty,x)\) are extreme value distributions. Therefore, without loss of generality we can assume \[ G(x,\infty)= \exp\{-(1+\gamma_1 x)^{- 1/\gamma_1}\}, \qquad G(\infty,y)= \exp\{-(1+\gamma_2 y)^{-1/\gamma_2}\} \] for some \(\gamma_1,\gamma_2\in \mathbb{R}\) and \(1+\gamma_1x>0\), \(1+\gamma_2y>0\). We are interested in rates of convergence in the limit relation. Under a second order regular variation condition, rates of convergence of the distribution of bivariate extreme order statistics to its limit distribution are given both in the total variation metric and in the uniform metric.
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 1040617
zbMATH Keywords: second order regular variation
zbMATH Keywords: extremes
author: Laurens De Haan
author: Liang Peng
MaRDI profile type: Publication
full work available at URL: jmva.1997.1669
OpenAlex ID: W2006790585
cites work: Q4726487
cites work: Q3762535
cites work: Second-order regular variation and rates of convergence in extreme-value theory
cites work: Approximation rates for multivariate exceedances
cites work: Rates of convergence in multivariate extreme value theory
cites work: Approximate distributions of order statistics. With applications to nonparametric statistics
cites work: Q3771297
DOI: 10.1006/JMVA.1997.1669
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description: scientific article; zbMATH DE number 1040617