[Forum SIS] Seminario Prof. M. Hallin

[Claudio Agostinelli]

Dipartimento di Matematica
Universita' di Trento

SEMINARIO

Prof. Marc Hallin
(ECARES and Departement de Mathematique Universite libre de Bruxelles)

MONGE-KANTOROVICH RANKS AND SIGNS

Data: 24 maggio 2016 Ore 14.30
Luogo: Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Aula
Seminari

Abstract: Unlike the real line, the real space R^K, K > 1 is not
"naturally" ordered. As a consequence, such fundamentals univariate
concepts as quantile and distribution functions, ranks, signs, all
order-related, do not straightforwardly extend to the multivariate
context. Since no universal pre-existing order exists, each
distribution, each data set, has to generate its own—the rankings behind
sensible concepts of multivariate quantile, ranks, or signs, inherently
will be distribution-specific and, in empirical situations, data-driven.
Many proposals have been made in the literature for such orderings—all
extending some aspects of the univariate concepts, but failing to
preserve the essential properties that make classical rank-based
inference a major inferential tool in the analysis of semiparametric
models where the density of some underlying noise remains unspecified:
(i) exact distribution-freeness, and (ii) asymptotic semiparametric
efficiency, see Hallin and Werker (2003).
Ranks and signs, and the resulting inference methods, are well
understood and well developed, essentially, in two cases:
one-dimensional observations, and elliptically symmetric ones. We start
by establishing the close connection, in those two cases, between
classical ranks and signs and measure transportation results, showing
that the rank transformation there actually reduces to an empirical
version of the unique gradient of convex function mapping a distribution
to the uniform over the unit ball. That fact, along with a result by
McCann (1995), itself extending the celebrated polar factorization
Theorem by Brenier (1991), is then exploited to define fully general
concepts of ranks and signs - called the Monge-Kantorovich ranks and
signs coinciding, in the univariate and elliptical settings, with the
traditional concepts, and enjoying under completely unspecified
(absolutely continuous) d-dimensional distributions, the essential
properties that make traditional rank-based inference an essential part
of the semiparametric inference.

Based on joint work with Victor Chernozhukov, Alfred Galichon, and Marc
Henry

Referente: Claudio Agostinelli

Tutti gli interessati sono cordialmente invitati a partecipare.

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Claudio Agostinelli
Dipartimento di Matematica
Universita' di Trento
email: claudio.agostinelli at unitn.it
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