Victor Chernozhukov, Alfred Galichon, Marc Hallin, Marc Henry
arXiv 29 Dec 2014 · Mathematics — Statistics Theory · publishedThe Annals of Statistics (2017) · 154 citations (OpenAlex)
arXiv:1412.8434 · PDF · DOI · OpenAlex · Extracted main text
We propose new concepts of statistical depth, multivariate quantiles, 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 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 and signs, and show their consistency by establishing a uniform convergence property for empirical transport maps, which is of independent interest.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Villani, C (2003) Topics in Optimal Transportation | 0.851 | 13 | 4 | 62% |
| 2 | Liu, R. Y (1990) On a notion of data depth based on random simplices, The Annals of Statistics 18, 405–-414 | 0.737 | 3 | 2 | 100% |
| 3 | Zuo, Y., and Serfling, R (2000) General notions of statistical depth function, The Annals of Statistics 28, 461–-482 | 0.737 | 3 | 2 | 100% |
| 4 | Ekeland, I., Galichon, A., and Henry, M (2012) Comonotonic measures of multivariate risks, Mathematical Finance 22, 109–132 self | 0.644 | 2 | 2 | 100% |
| 5 | Hallin, M., and Paindaveine, D (2002) Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks, The Annals of Statistics 30, 1103… self | 0.644 | 2 | 2 | 100% |
| 6 | Koshevoy, G (2002) The Tukey depth characterizes the atomic measure, Journal of Multivariate Analysis 83, 360-–364 | 0.644 | 2 | 2 | 100% |
| 7 | McCann, R. J (1995) Existence and uniqueness of monotone measure-preserving maps | 0.644 | 2 | 2 | 100% |
| 8 | Tukey, J. W (1975) Mathematics and the picturing of data, in Proceedings of the International Congress of Mathematicians (Vancouver, B | 0.644 | 2 | 2 | 100% |
| 9 | Brenier, Y (1991) Polar factorization and monotone rearrangement of vector-valued functions, Communications in Pure and Applied Mathematics 44, 37… | 0.644 | 2 | 2 | 100% |
| 10 | van der Vaart, A. W., and Wellner, J. A (1996) Weak Convergence | 0.511 | 2 | 2 | 50% |
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