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Asymptotic behaviour of extreme geometric quantiles and their estimation under moment conditions

Stéphane Girard 1 Gilles Stupfler 2 
1 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology
Abstract : A popular way to study the tail of a distribution is to consider its extreme quantiles. While this is a standard procedure for univariate distributions, it is harder for multivariate ones, primarily because there is no universally accepted definition of what a multivariate quantile should be. In this paper, we focus on extreme geometric quantiles. Their asymptotics are established, both in direction and magnitude, under suitable moment conditions, when the norm of the associated index vector tends to one. In particular, it appears that if a random vector has a finite covariance matrix, then the magnitude of its extreme geometric quantiles grows at a fixed rate. We take advantage of these results to define an estimator of extreme geometric quantiles of such a random vector. The consistency and asymptotic normality of the estimator are established and our results are illustrated on some numerical examples.
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Preprints, Working Papers, ...
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Submitted on : Thursday, September 4, 2014 - 5:06:48 PM
Last modification on : Saturday, June 25, 2022 - 7:40:01 PM
Long-term archiving on: : Friday, December 5, 2014 - 10:41:46 AM


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  • HAL Id : hal-01060985, version 1


Stéphane Girard, Gilles Stupfler. Asymptotic behaviour of extreme geometric quantiles and their estimation under moment conditions. 2014. ⟨hal-01060985⟩



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