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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:TALK105877AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/105877
DESCRIPTION:Unlike the real line\, the d-dimensional space R^d
\, for d >\; 1\, is not canonically ordered. As
a consequence\, such fundamental and strongly orde
r-related univariate concepts as quantile and dist
ribution functions\, and their empirical counterpa
rts\, involving ranks and signs\, do not canonical
ly extend to the multivariate context. Palliating
that lack of a canonical ordering has remained an
open problem for more than half a century\, and ha
s generated an abundant literature\, motivating\,
among others\, the development of statistical dept
h and copula-based methods. We show here that\, un
like the many definitions that have been proposed
in the literature\, the measure transportation-bas
ed ones \; \; introduced in Chernozhukov e
t al. (2017) enjoy all the properties (distributio
n-freeness and preservation of semiparametric effi
ciency) that make univariate quantiles and ranks s
uccessful tools for semiparametric statistical inf
erence. We therefore propose a new center-outward
definition of multivariate distribution and quanti
le functions\, along with their empirical counterp
arts\, 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 Chernozh
ukov et al. (2017) (which assumes compact supports
or finite second-order moments)\, does not requir
e any moment assumptions. The \; resulting ran
ks 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 ord
er-preserving transformations) generating the fami
ly of absolutely continuous distributions\; this\,
in view of a general result by Hallin and Werker
(2003)\, implies preservation of semiparametric ef
ficiency. As for the resulting quantiles\, they ar
e equivariant under the same transformations\, whi
ch confirms the order-preserving nature of gradien
ts of convex function.

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