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Extremes

, Volume 18, Issue 4, pp 629–663 | Cite as

Extreme geometric quantiles in a multivariate regular variation framework

  • Stéphane Girard
  • Gilles Stupfler
Article

Abstract

Considering extreme quantiles is a popular way to understand the tail of a distribution. While they have been extensively studied for univariate distributions, much less has been done for multivariate ones, primarily because there is no universally accepted definition of what a multivariate quantile or a multivariate distribution tail should be. In this paper, we focus on extreme geometric quantiles. In Girard and Stupfler (2015) Intriguing properties of extreme geometric quantiles, their asymptotics are established, both in direction and magnitude, under suitable integrability conditions, when the norm of the associated index vector tends to one. In this paper, we study extreme geometric quantiles when the integrability conditions are not fulfilled, in a framework of regular variation.

Keywords

Extreme quantile Geometric quantile Asymptotic behavior Multivariate regular variation 

AMS 2000 Subject Classifications

62H05 62G32 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Team Mistis, Inria Grenoble Rhône-Alpes & LJK, InovalléeMontbonnotFrance
  2. 2.Aix Marseille Université, CNRS, EHESS, Centrale MarseilleMarseilleFrance

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