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SUMMARY:Multivariate Distribution and Quantile Functions\, Ranks and Signs
 : A measure transportation approach - Marc Hallin (Université Libre de Br
 uxelles)
DTSTART:20180522T100000Z
DTEND:20180522T110000Z
UID:TALK107182@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Unlike the real line\, the d-dimensional space R^d\, for d > 1
 \, is not canonically ordered. As a consequence\, such fundamental and str
 ongly order-related univariate concepts as quantile and distribution funct
 ions\, and their empirical counterparts\, involving ranks and signs\, do n
 ot canonically extend to the multivariate context. Palliating that lack of
  a canonical ordering has remained an open problem for more than half a ce
 ntury\, and has generated an abundant literature\, motivating\, among othe
 rs\, the development of statistical depth and copula-based methods. We sho
 w here that\, unlike the many definitions that have been proposed in the l
 iterature\, the measure transportation-based ones&nbsp\;&nbsp\; introduced
  in Chernozhukov et al. (2017) enjoy all the properties (distribution-free
 ness and preservation of semiparametric efficiency) that make univariate q
 uantiles and ranks successful tools for semiparametric statistical inferen
 ce. We therefore propose a new center-outward definition of multivariate d
 istribution and quantile functions\, along with their empirical counterpar
 ts\, for which we establish a Glivenko-Cantelli result. Our approach\, bas
 ed on results by McCann (1995)\, is geometric rather than analytical and\,
  contrary to the Monge-Kantorovich one in Chernozhukov et al. (2017) (whic
 h assumes compact supports or finite second-order moments)\, does not requ
 ire any moment assumptions. The&nbsp\; 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 th
 us are playing the role of order-preserving transformations) generating th
 e family of absolutely continuous distributions\; this\, in view of a gene
 ral result by Hallin and Werker (2003)\, implies preservation of semiparam
 etric efficiency. As for the resulting quantiles\, they are equivariant un
 der the same transformations\, which confirms the order-preserving nature 
 of gradients of convex function. <br><br><br><br>
LOCATION:Seminar Room 2\, Newton Institute
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