EconPapers    
Economics at your fingertips  
 

On Distribution and Quantile Functions, Ranks and Signs in R_d

Marc Hallin ()

No ECARES 2017-34, Working Papers ECARES from ULB -- Universite Libre de Bruxelles

Abstract: Unlike the real line, the d-dimensional space Rd, for d ≥ 2, is not canonically ordered. As a consequence, such fundamental and strongly order-related 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 remained an open problem for more than half a century, and has generated an abundant literature, motivating, among others, the development of statistical depth and copula-based methods. We show here that, unlike the many definitions that have been proposed in the literature, the measure transportation-based ones introduced in Chernozhukov et al. (2017) enjoy all the properties (distribution-freeness and preservation of semiparametric efficiency) that make univariate quantiles and ranks successful tools for semiparametric statistical inference. We therefore 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, based on results by McCann (1995), is geometric rather than analytical and, contrary to the Monge-Kantorovich one in Chernozhukov et al. (2017) (which assumes compact supports or finite second-order moments), does not require any moment assumptions. The resulting ranks and signs are shown to be strictly distribution-free, and maximal invariant under the action of transformations (namely, the gradients of convex functions, which thus are playing the role of order-preserving transformations) generating the family of absolutely continuous distributions; this, in view of a general result by Hallin and Werker (2003), implies preservation of semiparametric efficiency. The resulting quantiles are equivariant under the same transformations, which confirms the order-preserving nature of gradients of convex function.

Keywords: multivariate distribution function; multivariate quantiles; multivariate ranks; multivariate signs; multivariate order-preserving transformation; glivenko-cantelli; invariance/equivariance; gradient of convex function (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-ecm
Date: 2017-09
References: Add references at CitEc
Citations: View citations in EconPapers (1) Track citations by RSS feed

Published by:

Downloads: (external link)
https://dipot.ulb.ac.be/dspace/bitstream/2013/2582 ... N-ondistribution.pdf Full text for the whole work, or for a work part (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eca:wpaper:2013/258262

Ordering information: This working paper can be ordered from
http://hdl.handle.ne ... lb.ac.be:2013/258262

Access Statistics for this paper

More papers in Working Papers ECARES from ULB -- Universite Libre de Bruxelles Contact information at EDIRC.
Bibliographic data for series maintained by Benoit Pauwels ().

 
Page updated 2019-06-24
Handle: RePEc:eca:wpaper:2013/258262