On the Behavior of Extreme d-dimensional Spatial Quantiles Under Minimal Assumptions

Abstract

Spatial or geometric quantiles are the only multivariate quantiles coping with both high-dimensional data and functional data, also in the framework of multiple-output quantile regression. This work studies spatial quantiles in the finitedimensional case, where the spatial quantile µ α,u (P) of the distribution P taking values in R d is a point in R d indexed by an order α ∈ [0, 1) and a direction u in the unit sphere S d−1 of R d -or equivalently by a vector αu in the open unit ball of R d . Recently, [13] proved that (i) the extreme quantiles µ α,u (P) obtained as α → 1 exit all compact sets of R d 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, which makes the proofs very challenging. We also weaken the well-known sufficient condition for uniqueness of finite-dimensional spatial quantiles.

From Geometric Quantiles to Halfspace Depths: A Geometric Approach for External Behaviour.

“…Finally, let us mention some recent results obtained in (Paindaveine and Virta, 2021, Theorems 1 & 2), which set the stage for our results in Section 3.1 when turning to sample quantiles.…”

“…Drawbacks on the characterisation of the tail behaviour of a distribution via the corresponding geometric quantiles were pointed in Girard and Stupfler (2015), showing a similar behaviour for extreme quantiles for distributions sharing the same covariance matrix. Some improvement has been provided in Paindaveine and Virta (2021) by considering the joint asymptotics of the sample size and the α-level of the geometrical quantile. Nevertheless, although very useful, these existing results fall short of being readily applicable as they are established for non-atomic measures.…”

“…As can be observed immediately from the definition of sample geometric quantiles, the underlying measure does not satisfy the smoothness properties needed to prove the results stated in the previous section. Therefore, we begin with elementary asymptotic estimates in Theorem 3.2, which can be seen as improvements of Theorems 2 & 3 of Paindaveine and Virta (2021). The proof of Theorem 3.2 is developed in Section 3.2.…”

“…Numerous results are already available for the population-based analysis of these geometric measures, particularly in terms of their asymptotic (extreme) behaviour. For geometric quantiles, refer to Girard and Stupfler (2017), Paindaveine and Virta (2021). For halfspace depth, assuming multivariate regularly varying distributions, the work by He and Einmahl (2017) is noteworthy.…”

“…Concerning geometric quantiles, proofs are developed with classical tools of probability, extending results by Girard and Stupfler (2017) in the population setting, and using results by Chaudhury (1996), Paindaveine and Virta (2021) and the Glivenko-Cantelli theorem for the sample version.…”