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

By Marc Hallin, Eustasio del Barrio, Juan Antonio Cuesta-Albertos, Carlos Matrán. Annals of Statistics, 2021

160 citations recorded by the source corpus.

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 copulabased methods. We show that, unlike the many definitions proposed in the literature, the measure transportation-based ranks introduced in Chernozhukov et al. (2017) 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 (1995) and our results, unlike those of Chernozhukov et al. (2017), 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 (1959) 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, is provided. A numerical study of the resulting empirical quantile contours is conducted. MSC 2010 subject classifications: Primary 62G30; secondary 62B05.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4208a036356bb91dd950b499c079a474acf35356

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Citations in the corpus

Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs (2019, background): “” It has been shown that under the existence of finite marginal first moments, the distance covariance is zero if and only if H0 holds. For further extensions, Lyons (2013) generalized distance covariance/correlation to general metric spaces, and Jakobsen (2017) considered the corresponding test of ”
Choosing Among Notions of Multivariate Depth Statistics (2020, background): “They give rise to distribution free notions of multivariate quantiles, ranks, and signs (Hallin et al., 2021).”
Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation (2019, methodology): “In the d-dimensional Euclidean space, for d ≥ 2, due to the absence of a canonical ordering, the existing extensions of concepts like ranks (such as component-wise ranks, e.g., Bickel 1965; Puri and Sen 1966; spatial ranks, e.g., Chaudhuri 1996; Marden 1999; depth-based ranks, e.g., Liu and Singh 19”
Stability of entropic optimal transport and Schrödinger bridges (2021, methodology): “Multivariate ranks have been defined in nonparametric statistics through Brenier’s optimal transport map to extend the usual scalar notions and tests; see [15, 19, 20, 28].”
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).”