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On the estimation of a star-shaped set

Part of: Geometric probability and stochastic geometry Nonparametric inference

Published online by Cambridge University Press:  01 July 2016

Amparo Baíllo  and
Antonio Cuevas
Amparo Baíllo*
Affiliation:
Universidad Autónoma de Madrid
Antonio Cuevas*
Affiliation:
Universidad Autónoma de Madrid
*
∗ Postal address: Departamento de Matemáticas, Facultad de Ciencias, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain.
∗ Postal address: Departamento de Matemáticas, Facultad de Ciencias, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain.
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Abstract

The estimation of a star-shaped set S from a random sample of points X1,…,Xn ∊ S is considered. We show that S can be consistently approximated (with respect to both the Hausdorff metric and the ‘distance in measure’ between sets) by an estimator ŝn defined as a union of balls centered at the sample points with a common radius which can be chosen in such a way that ŝn is also star-shaped. We also prove that, under some mild conditions, the topological boundary of the estimator ŝn converges, in the Hausdorff sense, to that of S; this has a particular interest when the proposed estimation problem is considered from the point of view of statistical image analysis.

Keywords

Star-shaped setsset estimationHausdorff metricboundary convergence

MSC classification

Primary: 62G05: Estimation 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 4 , December 2001 , pp. 717 - 726
Copyright
Copyright © Applied Probability Trust 2001 

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