Rates of uniform convergence of extreme order statistics
type: Item
instance of: scholarly article
title: Rates of uniform convergence of extreme order statistics
zbMATH Open document ID: 0605.62045
DOI: 10.1007/BF02482514
author: Michael Falk
published in: Annals of the Institute of Statistical Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The main result of the paper is a sharp upper bound of the variational distance between the distribution of an extreme order statistic and an extreme value distribution. The concept of generalized Pareto distributions is crucial for the derivation and the formulation of the upper bound. Let f be the underlying density and \(w_ n\) the density of a shifted generalized Pareto distribution which corresponds to the approximating extreme value distribution G. Then the upper bound of the remainder term of the approximation is of the form \[ O[\int (\frac{a_ nf(a_ nx+b_ n)}{w_ n}-1)^ 2dG]^{1/2} \] where \(a_ n>0\) and the real \(b_ n\) are normalizing constants. Of equal importance is the extension of the one-dimensional result to the joint distribution of extreme order statistics. Several examples highlight the applicability of the results mentioned above.
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 3978154
zbMATH Keywords: rates of uniform convergence
zbMATH Keywords: joint distribution of extremes
zbMATH Keywords: normal extremes
zbMATH Keywords: sharp upper bound
zbMATH Keywords: variational distance
zbMATH Keywords: extreme order statistic
zbMATH Keywords: extreme value distribution
zbMATH Keywords: generalized Pareto distributions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3978154