STS | Prof. Marc Hallin | Multivariate Distribution and Quantile Functions, Ranks and Signs

Описание: STS | Prof. Marc Hallin | Multivariate Distribution and Quantile Functions, Ranks and Signs: A measure transportation approach

Speaker: Professor Marc Hallin (Université Libre de Bruxelles)
Date: 22nd May 2018 - 11:00 to 12:00
Venue: INI Seminar Room 2
Title: Multivariate Distribution and Quantile Functions, Ranks and Signs: A measure transportation approach
Event: (STS) Statistical scalability
Abstract: Unlike the real line,
the d-dimensional space R^d, for d > 1, 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 preservation 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 compact
supports or finite second-order moments), 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. As for the resulting
quantiles, they are equivariant under the same transformations, which confirms
the order-preserving nature of gradients of convex function.

-------------------

FOLLOW US
🌐| Website: https://www.newton.ac.uk
🎥| Main Channel:    / @isaacnewtoninstitute  
🐦| Twitter:   / newtoninstitute  
💬| Facebook:   / newton.institute  
📷| Instagram:   / isaacnewtoninstitute  
🔗| LinkedIn:   / isaac-newton-institute-for-mathematical-sc...  

SEMINAR ROOMS
🥇| INI Seminar Room 1:    / @iniseminarroom1  
🥈| INI Seminar Room 2:    / @iniseminarroom2  
🛰️| INI Satellite Events:    / @inisatellite  

ABOUT
The Isaac Newton Institute is a national and international visitor research institute. It runs research programmes on selected themes in mathematics and the mathematical sciences with applications over a wide range of science and technology. It attracts leading mathematical scientists from the UK and overseas to interact in research over an extended period.

👉 Learn more about us and our events here: https://www.newton.ac.uk