Indian Statistical Institute

P.C. Mahalanobis Memorial Lectures

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

by

Professor Marc Hallin

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.

Lecture

Date      :      13th February, 2018 (Tuesday)

Time      :      3.30 pm

Title       :      On Multivariate Distribution  and

Quantile Functions, Ranks and Signs: a Measure Transportation Approach

Abstract

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.

The lecture will be held in Platinum Jubilee Auditorium.

All are welcome.