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On the behavior of extreme d-dimensional spatial quantiles under minimal assumptions

par Paindaveine, Davy ;Virta, Joni
Editeur scientifique Ruiz-Gazen, Anne;Daouia, Abdelaati
Référence On the behavior of extreme d-dimensional spatial quantiles under minimal assumptions, Springer, Cham., Vol. Advances in Contemporary Statistics and Econometrics, Advances in Contemporary Statistics and Econometrics
Publication Publié, 2021

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Résumé :

Spatial or geometric quantiles are among the most celebrated concepts of multivariate quantiles. The spatial quantile μα,u(P) of a probability measure P over Rd is a point in Rd indexed by an order α ∈ [0, 1) and a direction u in the unit sphere Sd-1 of Rd-or equivalently by a vector αu in the open unit ball of Rd. Recently, Girard and Stupfler (2017) proved that (i) the extreme quantiles μα,u(P) obtained as α → 1 exit all compact sets of Rd and that (ii) they do so in a direction converging to u. These results help understanding the nature of these quantiles: the first result is particularly striking as it holds even if P has a bounded support, whereas the second one clarifies the delicate dependence of spatial quantiles on u. However, they were established under assumptions imposing that P is non-atomic, so that it is unclear whether they hold for empirical probability measures. We improve on this by proving these results under much milder conditions, allowing for the sample case. This prevents using gradient condition arguments, whichmakes the proofs very challenging.We also weaken the well-known sufficient condition for the uniqueness of finite-dimensional spatial quantiles.