MONGE-KANTOROVICH DEPTH, QUANTILES, RANKS AND SIGNS

• 成果类型: Article
• 署名作者: Chernozhukov, Victor; Galichon, Alfred; Hallin, Marc; Henry, Marc
• 署名单位: Massachusetts Institute of Technology (MIT); Massachusetts Institute of Technology (MIT); New York University; New York University; Institut d'Etudes Politiques Paris (Sciences Po); Universite Libre de Bruxelles; Pennsylvania Commonwealth System of Higher Education (PCSHE); Pennsylvania State University; Pennsylvania State University - University Park
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/16-AOS1450
• 发表日期: 2017
• 页码: 223-256
• 关键词: multivariate-analysis; optimal tests; tukey depth; regression; inference; notion; shape

摘要

We propose new concepts of statistical depth, multivariate quantiles, vector quantiles and ranks, ranks and signs, based on canonical transportation maps between a distribution of interest on R-d and a reference distribution on the d-dimensional unit ball. The new depth concept, called Monge Kantorovich depth, specializes to halfspace depth for d = 1 and in the case of spherical distributions, but for more general distributions, differs from the latter in the ability for its contours to account for non-convex features of the distribution of interest. We propose empirical counterparts to the population versions of those Monge Kantorovich depth contours, quantiles, ranks, signs and vector quantiles and ranks, and show their consistency by establishing a uniform convergence property for empirical (forward and reverse) transport maps, which is the main theoretical result of this paper.