Monge–Kantorovich depth, quantiles, ranks and signs
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- Kantoro...
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| Format: | Article |
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Institute of Mathematical Statistics
2018
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| Online Access: | http://hdl.handle.net/1721.1/113890 https://orcid.org/0000-0002-3250-6714 |
| _version_ | 1826196471429464064 |
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| author | Galichon, Alfred Hallin, Marc Henry, Marc Chernozhukov, Victor V |
| author2 | Massachusetts Institute of Technology. Department of Economics |
| author_facet | Massachusetts Institute of Technology. Department of Economics Galichon, Alfred Hallin, Marc Henry, Marc Chernozhukov, Victor V |
| author_sort | Galichon, Alfred |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T10:27:23Z |
| format | Article |
| id | mit-1721.1/113890 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T10:27:23Z |
| publishDate | 2018 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/1138902022-09-30T21:17:08Z Monge–Kantorovich depth, quantiles, ranks and signs Galichon, Alfred Hallin, Marc Henry, Marc Chernozhukov, Victor V Massachusetts Institute of Technology. Department of Economics Chernozhukov, Victor V 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. National Science Foundation (U.S.) (SES 1061841) 2018-02-26T20:07:41Z 2018-02-26T20:07:41Z 2017-02 2018-02-20T17:33:31Z Article http://purl.org/eprint/type/JournalArticle 0090-5364 http://hdl.handle.net/1721.1/113890 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. https://orcid.org/0000-0002-3250-6714 http://dx.doi.org/10.1214/16-AOS1450 The Annals of Statistics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Institute of Mathematical Statistics arXiv |
| spellingShingle | Galichon, Alfred Hallin, Marc Henry, Marc Chernozhukov, Victor V Monge–Kantorovich depth, quantiles, ranks and signs |
| title | Monge–Kantorovich depth, quantiles, ranks and signs |
| title_full | Monge–Kantorovich depth, quantiles, ranks and signs |
| title_fullStr | Monge–Kantorovich depth, quantiles, ranks and signs |
| title_full_unstemmed | Monge–Kantorovich depth, quantiles, ranks and signs |
| title_short | Monge–Kantorovich depth, quantiles, ranks and signs |
| title_sort | monge kantorovich depth quantiles ranks and signs |
| url | http://hdl.handle.net/1721.1/113890 https://orcid.org/0000-0002-3250-6714 |
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