Convergence of Multivariate Quantile Surfaces

Abstract

We define the quantile set of order $\alpha \in \left[1/2,1\right)$ associated to a law $P$ on ${\mathbb{R}}^d$ to be the collection of its directional quantiles seen from an observer $O\in {\mathbb{R}}^d$. Under minimal assumptions these star-shaped sets are closed surfaces, continuous in $(O,\alpha )$ and the collection of empirical quantile surfaces is uniformly consistent.\ Under mild assumptions -- no density or symmetry is required for $P$ -- our uniform central limit theorem reveals the correlations between quantile points and a non asymptotic Gaussian approximation provides joint confident enlarged quantile surfaces. Our main result is a dimension free rate $n^{-1/4}(\log n{)}^{1/2}(\log \log n{)}^{1/4}$ of Bahadur-Kiefer embedding by the empirical process indexed by half-spaces. These limit theorems sharply generalize the univariate quantile convergences and fully characterize the joint behavior of Tukey half-spaces.

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