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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Multivariate Distribution and Quantile Functions\,
Ranks and Signs: A measure transportation approac
h - Marc Hallin (Université Libre de Bruxelles)
DTSTART;TZID=Europe/London:20180522T110000
DTEND;TZID=Europe/London:20180522T120000
UID:TALK107182AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/107182
DESCRIPTION:Unlike the real line\, the d-dimensional space R^d
\, for d > 1\, is not canonically ordered. As a co
nsequence\, such fundamental and strongly order-re
lated univariate concepts as quantile and distribu
tion functions\, and their empirical counterparts\
, involving ranks and signs\, do not canonically e
xtend to the multivariate context. Palliating that
lack of a canonical ordering has remained an open
problem for more than half a century\, and has ge
nerated an abundant literature\, motivating\, amon
g others\, the development of statistical depth an
d copula-based methods. We show here that\, unlike
the many definitions that have been proposed in t
he literature\, the measure transportation-based o
nes \; \; introduced in Chernozhukov et al
. (2017) enjoy all the properties (distribution-fr
eeness and preservation of semiparametric efficien
cy) that make univariate quantiles and ranks succe
ssful tools for semiparametric statistical inferen
ce. We therefore propose a new center-outward defi
nition of multivariate distribution and quantile f
unctions\, along with their empirical counterparts
\, for which we establish a Glivenko-Cantelli resu
lt. Our approach\, based on results by McCann (199
5)\, is geometric rather than analytical and\, con
trary to the Monge-Kantorovich one in Chernozhukov
et al. (2017) (which assumes compact supports or
finite second-order moments)\, does not require an
y moment assumptions. The \; resulting ranks a
nd signs are shown to be strictly distribution-fre
e\, and maximal invariant under the action of tran
sformations (namely\, the gradients of convex func
tions\, which thus are playing the role of order-p
reserving transformations) generating the family o
f absolutely continuous distributions\; this\, in
view of a general result by Hallin and Werker (200
3)\, implies preservation of semiparametric effici
ency. As for the resulting quantiles\, they are eq
uivariant under the same transformations\, which c
onfirms the order-preserving nature of gradients o
f convex function.

LOCATION:Seminar Room 2\, Newton Institute
CONTACT:INI IT
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