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Towards a universally consistent estimatorof the Minkowski content∗

Published online by Cambridge University Press:  17 May 2013

Antonio Cuevas ,
Ricardo Fraiman  and
László Györfi
Antonio Cuevas
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain. [email protected]
Ricardo Fraiman
Affiliation:
Departamento de Matemáticas y Ciencias, Universidad de San Andrés, Argentina and CMAT, Universidad de la República, Uruguay
László Györfi
Affiliation:
Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Hungary
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Abstract

We deal with a subject in the interplay between nonparametric statistics and geometricmeasure theory. The measure L0(G) of theboundary of a set G ⊂ ℝd (withd ≥ 2) can be formally defined, via a simple limit, bythe so-called Minkowski content. We study the estimation ofL0(G) from a sample of random points insideand outside G. The sample design assumes that, for each sample point, weknow (without error) whether or not that point belongs to G. Under thisdesign we suggest a simple nonparametric estimator and investigate its consistencyproperties. The main emphasis in this paper is on generality. So we are especiallyconcerned with proving the consistency of our estimator under minimal assumptions on theset G. In particular, we establish a mild shape condition onG under which the proposed estimator is consistent inL2. Roughly speaking, such condition establishes that theset of “very spiky” points at the boundary of G must be “small”. This isformalized in terms of the Minkowski content of such set. Several examples arediscussed.

Keywords

Minkowski contentnonparametric set estimationboundary estimation
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 359 - 369
Copyright
© EDP Sciences, SMAI, 2013

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