Indian Statistical Institute

*On Multivariate Distribution and Quantile Functions, Ranks and Signs: a Measure
Transportation Approach*

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**Marc Hallin holds a PhD in Sciences &
Mathematics from the Universite libre
de Bruxelles (1976). He is co-Editor-in-Chief of
Statistical Inference for Stochastic Processes and an Associate Editor of the
Journal of the American Statistical Association, the Journal of Econometrics,
the Annals of Computational and Financial Econometrics, the Journal of the
Japan Statistical Society, and the Annales de l'Institut de Statistique de l'Universite de Paris. A Fellow of the Institute of
Mathematical Statistics (I.M.S.), of the American Statistical Association
(A.S.A.), and of the International Statistical Institute (I.S.I.), he is member
of the Classe des Sciences of the Royal Academy of
Belgium.**

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**Date
: 13th February, 2018 (Tuesday)**

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**Time
: 3.30 pm**

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**Title
: On Multivariate Distribution
and **

**Quantile**** Functions, Ranks and
Signs: a Measure Transportation Approach**

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__Abstract__

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**Unlike the
real line, the ****-dimensional space ,
for ****,
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 reservation 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 compactly supported distributions), 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.**

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**The lecture will be held in Platinum Jubilee Auditorium.**

**All are welcome****.**

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