2016/07/09 by Adil Ahidar-Coutrix, Ahidar-Coutrix, Adil, Philippe Berthet +1
Computer Science · Mathematics · #Applied mathematics #Bayesian Methods and Mixture Models #Central limit theorem #Combinatorics #FOS: Mathematics #Gaussian #Limit (mathematics) #Mathematical analysis #Mathematics #Multivariate statistics #Quantile #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics #Statistics Theory (math.ST) #Univariate #Weak convergence #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1607.02604
version 2
openalex publication_date 2016/07/09 · arxiv created 2016/12/05 · arxiv updated 2016/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define the quantile set of order α∈ [ 1/2,1) associated to a law P on ℝd to be the collection of its directional quantiles seen from an observer O∈ ℝd. Under minimal assumptions these star-shaped sets are closed surfaces, continuous in (O,α) 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(loglog 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.