Monge-Kantorovich Depth, Quantiles, Ranks and Signs

By Victor Chernozhukov, Alfred Galichon, Marc Hallin, Marc Henry. 2014

283 citations recorded by the source corpus.

We propose new concepts of statistical depth, multivariate quantiles,ranks and signs, based on canonical transportation maps between a distributionof interest on IRd and a reference distribution on the d-dimensionalunit ball. The new depth concept, called Monge-Kantorovich depth, specializesto halfspace depth in the case of elliptical distributions, but, for more generaldistributions, differs from the latter in the ability for its contours to account fornon convex features of the distribution of interest. We propose empirical counterpartsto the population versions of those Monge-Kantorovich depth contours,quantiles, ranks and signs, and show their consistency by establishing a uniformconvergence property for transport maps, which is of independent interest.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/569df88e3d75a6373d01a2430c70f9869c3f8d87

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Statistical Aspects of Wasserstein Distances (2018, background): “Chernozhukov et al. (2017) describe the resulting form of the quantile contours and the induced notions of depth and ranks, and estimate them from data.”
Multivariate goodness-of-fit tests based on Wasserstein distance (2020, background): “In a recent strand of literature, measure transportation serves to link a multivariate probability measure to a standard reference distribution, yielding novel concepts of multivariate ranks, signs, and quantiles (Carlier et al., 2016; Chernozhukov et al., 2017; Hallin et al., 2020).”
Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation (2019, background): “The theoretical results in [28] (including uniform convergence of rank maps) assume compactly supported distributions, whereas [44] proves a more general result under the condition that the population rank map is a homeomorphism — still a relatively strong assumption.”
Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach (2021, methodology): “Moment assumptions (as in Boeckel et al. (2018) or Chernozhukov et al. (2017) where (weak) consistency is established under compactly supported imsart-generic ver.”
On the continuity of center-outward distribution and quantile functions (2018, background): “Motivated by this lack of a statistically sound definition, first in [6] and then in [12], the authors proposed a new concept of center-outward distribution function based on optimal transportation ideas.”
An Optimal Transportation Approach for Assessing Almost Stochastic Order (2017, background): “…distance is deﬁned as In the univariate case, W 2 equals the L 2 -distance between quantile functions, namely, Statistical applications based on optimal transportation, and particularly on the L 2 version, are receiving considerable attention in recent times (see e.g. [8], [9], [10], [17] or [5]).”