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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:TALK105877@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Unlike the real line\, the d-dimensional space R^d\, for d &gt
 \; 1\, is not canonically ordered. As a consequence\, such fundamental and
  strongly order-related univariate concepts as quantile and distribution f
 unctions\, and their empirical counterparts\, involving ranks and signs\, 
 do not canonically extend to the multivariate context. Palliating that lac
 k 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 t
 he literature\, the measure transportation-based ones&nbsp\;&nbsp\; introd
 uced in Chernozhukov et al. (2017) enjoy all the properties (distribution-
 freeness and preservation of semiparametric efficiency) that make univaria
 te quantiles and ranks successful tools for semiparametric statistical inf
 erence. We therefore propose a new center-outward definition of multivaria
 te distribution and quantile functions\, along with their empirical counte
 rparts\, for which we establish a Glivenko-Cantelli result. Our approach\,
  based on results by McCann (1995)\, is geometric rather than analytical a
 nd\, 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&nbsp\; resulting ranks and signs are s
 hown to be strictly distribution-free\, and maximal invariant under the ac
 tion of transformations (namely\, the gradients of convex functions\, whic
 h thus are playing the role of order-preserving transformations) generatin
 g the family of absolutely continuous distributions\; this\, in view of a 
 general result by Hallin and Werker (2003)\, implies preservation of semip
 arametric efficiency. As for the resulting quantiles\, they are equivarian
 t under the same transformations\, which confirms the order-preserving nat
 ure of gradients of convex function. <br><br><br><br>
LOCATION:Seminar Room 2\, Newton Institute
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