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Towards a universally consistent estimator of 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. antonio.cuevas@uam.es
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 geometric measure theory. The measure L0(G) of the boundary of a set G ⊂ ℝd (with d ≥ 2) can be formally defined, via a simple limit, by the so-called Minkowski content. We study the estimation of L0(G) from a sample of random points inside and outside G. The sample design assumes that, for each sample point, we know (without error) whether or not that point belongs to G. Under this design we suggest a simple nonparametric estimator and investigate its consistency properties. The main emphasis in this paper is on generality. So we are especially concerned with proving the consistency of our estimator under minimal assumptions on the set G. In particular, we establish a mild shape condition on G under which the proposed estimator is consistent in L2. Roughly speaking, such condition establishes that the set of “very spiky” points at the boundary of G must be “small”. This is formalized in terms of the Minkowski content of such set. Several examples are discussed.

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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