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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.
Authors
Professor of Economics and Mathematics NYU
Marc Henry
Université Libre de Bruxelles
Working Paper details
- DOI
- 10.1920/wp.cem.2015.5715
- Publisher
- Institute for Fiscal Studies
Suggested citation
Chernozhukov, V et al. (2015). Monge-Kantorovich depth, quantiles, ranks and signs. London: Institute for Fiscal Studies. Available at: https://ifs.org.uk/publications/monge-kantorovich-depth-quantiles-ranks-and-signs-0 (accessed: 25 April 2024).
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