Extreme geometric quantiles in a multivariate regular variation framework
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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.
KeywordsExtreme quantile Geometric quantile Asymptotic behavior Multivariate regular variation
AMS 2000 Subject Classifications62H05 62G32
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- Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge, U.K. (1987)Google Scholar
- Geyer, C.J.: On the Asymptotics of Convex Stochastic Optimization, unpublished manuscript (1996)Google Scholar
- Girard, S., Stupfler, G.: Intriguing properties of extreme geometric quantiles, REVSTAT – Statistical Journal, to appear. Available at http://hal.inria.fr/hal-00865767(2015)
- de Haan, L., Ferreira, A.: Extreme value theory: an introduction. Springer, New York (2006)Google Scholar
- Kemperman, J.H.B.: The median of a finite measure on a Banach space. In: Y. Dodge (ed.) Statistical data analysis based on the L 1–norm and related methods, pp 217–230. Amsterdam, North Holland (1987)Google Scholar
- Knight, K.: Epi-convergence in Distribution and Stochastic Equi-semicontinuity, technical report , University of Toronto (1999)Google Scholar
- Resnick, S.I.: Heavy tail phenomena: Probabilistic and statistical modeling, Springer Series in Operations Research and Financial Engineering. Springer-Verlag, New York (2006)Google Scholar