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Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach

par Hallin, Marc ;del Barrio, Eustasio;Cuesta-Albertos, Juan;Matrán, Carlos
Référence Annals of statistics, 49, 2, page (1139-1165)
Publication Publié, 2021-04-01

Article révisé par les pairs

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Résumé :

Unlike the real line, the real space Rd, for d ≥ 2, 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.