April 2021 Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach
Marc Hallin, Eustasio del Barrio, Juan Cuesta-Albertos, Carlos Matrán
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Ann. Statist. 49(2): 1139-1165 (April 2021). DOI: 10.1214/20-AOS1996


Unlike the real line, the real space Rd, for d2, is not canonically ordered. As a consequence, such fundamental univariate concepts as quantile and distribution functions and their empirical counterparts, involving ranks and signs, do not canonically extend to the multivariate context. Palliating that lack of a canonical ordering has been an open problem for more than half a century, generating an abundant literature and motivating, among others, the development of statistical depth and copula-based methods. We show that, unlike the many definitions proposed in the literature, the measure transportation-based ranks and signs introduced in Chernozhukov, Galichon, Hallin and Henry (Ann. Statist. 45 (2017) 223–256) enjoy all the properties that make univariate ranks a successful tool for semiparametric inference. Related with those ranks, we propose a new center-outward definition of multivariate distribution and quantile functions, along with their empirical counterparts, for which we establish a Glivenko–Cantelli result. Our approach is based on McCann (Duke Math. J. 80 (1995) 309–323) and our results do not require any moment assumptions. The resulting ranks and signs are shown to be strictly distribution-free and essentially maximal ancillary in the sense of Basu (Sankhyā 21 (1959) 247–256) which, in semiparametric models involving noise with unspecified density, can be interpreted as a finite-sample form of semiparametric efficiency. Although constituting a sufficient summary of the sample, empirical center-outward distribution functions are defined at observed values only. A continuous extension to the entire d-dimensional space, yielding smooth empirical quantile contours and sign curves while preserving the essential monotonicity and Glivenko–Cantelli features of the concept, is provided. A numerical study of the resulting empirical quantile contours is conducted.


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Marc Hallin. Eustasio del Barrio. Juan Cuesta-Albertos. Carlos Matrán. "Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach." Ann. Statist. 49 (2) 1139 - 1165, April 2021. https://doi.org/10.1214/20-AOS1996


Received: 1 May 2019; Revised: 1 June 2020; Published: April 2021
First available in Project Euclid: 2 April 2021

Digital Object Identifier: 10.1214/20-AOS1996

Primary: 62G30
Secondary: 62B05

Keywords: ancillarity , Basu theorem , cyclical monotonicity , distribution-freeness , Glivenko–Cantelli theorem , multivariate distribution function , Multivariate quantiles , multivariate ranks , multivariate signs

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 2 • April 2021
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