Convergence of Multivariate Quantile Surfaces
| Autor: | Ahidar-Coutrix, Adil, Berthet, Philippe |
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| Rok vydání: | 2016 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: version 2 |
| Databáze: | arXiv |
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