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SUMMARY:The long quest for quantiles and ranks in $R^d$ and on manifolds -
  Marc Hallin (Université Libre de Bruxelles)
DTSTART:20250506T104500Z
DTEND:20250506T114500Z
UID:TALK230437@talks.cam.ac.uk
DESCRIPTION:Quantiles are a fundamental concept in probability\, and an es
 sential tool in statistics\, from descriptive to inferential. Still\, desp
 ite half a century of attempts\, no satisfactory and fully agreed-upon def
 inition of the concept\, and the "dual" notion of ranks\, is available bey
 ond the well-understood case of univariate variables and distributions. Th
 e need for such a definition is particularly critical for variables taking
  values in ${\\mathbb R}^d$\, for directional variables (values on the hyp
 ersphere)\, and\, more generally\, for variables with values on manifolds.
  Unlike the real line\, indeed\, no canonical ordering is available on the
 se domains. We show how measure transportation brings a solution to this p
 roblem by characterizing distribution-specific (data-driven\, in the empir
 ical case) orderings and {\\it center-outward} distribution and quantile f
 unctions (ranks and signs in the empirical case) that satisfy all the prop
 erties expected from such concepts while reducing\, in the case of real-va
 lued variables\, to the classical univariate notion.
LOCATION:Seminar Room 1\, Newton Institute
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