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Lundi 29 mai 2017

Intervenants : Nicolaï BALDIN (University of Cambridge)

«The wrapping hull and a unified framework for volume estimation »

De 15h à 16h15, en salle 08 à l'ENSAE : 3 avenue Pierre Larousse à Malakoff (Tram T3: "Porte de Vanves" ou Métro 13 : "Porte de Vanves" ou "Malakoff Plateau de Vanves")

 

In this talk, I am going to present a unified framework for estimating the volume of a set in $\mathbb{R}^d$ based on observations of points uniformly distributed over the set. The framework applies to all classes of sets satisfying one simple axiom: a class is assumed to be intersection stable. No further hypotheses on the boundary of the set are imposed; in particular, the convex sets and the so-called weakly-convex sets are covered by the framework. The approach rests upon a homogeneous Poisson point process model. We shall discuss the so-called wrapping hull, a generalization of the convex hull, and prove that it is a sufficient and complete statistic. The proposed estimator of the volume is shown to be consistent for all classes of sets satisfying the axiom and mimics an unbiased estimator with uniformly minimal variance. The construction and proofs rely on a beautiful interplay between probabilistic and geometric arguments. On the way, we shall encounter Poisson point processes, martingales and some open questions in stochastic geometry.

Ce séminaire est organisé par :

Alexandre TSYBAKOV         (Laboratoire de Statistique-CREST)

Cristina BUTUCEA                (Laboratoire de Statistique-CREST)