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On boundary estimation

Part of: Nonparametric inference Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Antonio Cuevas  and
Alberto Rodríguez-Casal
Antonio Cuevas*
Affiliation:
Universidad Autónoma de Madrid
Alberto Rodríguez-Casal*
Affiliation:
Universidad de Vigo
*
∗ Postal address: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain. Email address: antonio.cuevas@uam.es
∗∗ Postal address: Departamento de Estadística e Investigación Operativa, Universidad de Vigo, Facultad de Ciencias Económicas y Empresariales, 36200 Vigo, Spain
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Abstract

We consider the problem of estimating the boundary of a compact set S ⊂ ℝd from a random sample of points taken from S. We use the Devroye-Wise estimator which is a union of balls centred at the sample points with a common radius (the smoothing parameter in this problem). A universal consistency result, with respect to the Hausdorff metric, is proved and convergence rates are also obtained under broad intuitive conditions of a geometrical character. In particular, a shape condition on S, which we call expandability, plays an important role in our results. The simple structure of the considered estimator presents some practical advantages (for example, the computational identification of the boundary is very easy) and makes this problem quite close to some basic issues in stochastic geometry.

Keywords

Boundary estimationset estimationHausdorff metricexpandable setstandard set

MSC classification

Primary: 62G05: Estimation 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 36 , Issue 2 , June 2004 , pp. 340 - 354
Copyright
Copyright © Applied Probability Trust 2004 

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