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SUMMARY:The wrapping hull and a unified framework for estimating the volum
 e of a body - Nicolai Baldin (University of Cambridge)
DTSTART:20170315T160000Z
DTEND:20170315T170000Z
UID:TALK71600@talks.cam.ac.uk
CONTACT:Mo Dick Wong
DESCRIPTION:In this talk\, I am going to present a unified framework for e
 stimating the volume of a set in $\\mathbb{R}^d$ based on observations of 
 points uniformly distributed over the set.\nThe framework applies to all c
 lasses of sets satisfying one simple axiom: a class is assumed to be inter
 section stable. No further hypotheses on the boundary of the set are impos
 ed\; in particular\, the convex sets and the so-called weakly-convex sets 
 are covered by the framework. The approach rests upon a homogeneous Poisso
 n point process model. We introduce the so-called wrapping hull\, a genera
 lization of the convex hull\, and prove that it is a sufficient and comple
 te statistic. The proposed estimator of the volume is simply the volume of
  the wrapping hull scaled with an appropriate factor. It is shown to be co
 nsistent for all classes of sets satisfying the axiom and mimics an unbias
 ed estimator with uniformly minimal variance. The construction and proofs 
 rely on a beautiful interplay between probabilistic and geometric argument
 s. On the way\, we shall encounter Poisson point processes\, martingales\,
  hulls and new open problems in stochastic geometry.
LOCATION:MR4\, Centre for Mathematical Sciences
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