Monge–Kantorovich depth, quantiles, ranks and signs
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1412.8434.pdf
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Author(s)
Galichon, Alfred
Hallin, Marc
Henry, Marc
Chernozhukov, Victor V
Date Issued
February 2017
Journal
The Annals of Statistics
Publisher
Institute of Mathematical Statistics
Citation
Chernozhukov, Victor, Alfred Galichon, Marc Hallin, and Marc Henry. “Monge–Kantorovich Depth, Quantiles, Ranks and Signs.” The Annals of Statistics 45, no. 1 (February 2017): 223–256.
Version
Original manuscript
Abstract
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 Rd 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.
MIT Department
Massachusetts Institute of Technology. Department of Economics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1214/16-AOS1450
