Volume 17, 2013
|Page(s)||359 - 369|
|Published online||17 May 2013|
Towards a universally consistent estimator of the Minkowski content∗
Departamento de Matemáticas, Universidad Autónoma de
2 Departamento de Matemáticas y Ciencias, Universidad de San Andrés, Argentina and CMAT, Universidad de la República, Uruguay
3 Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Hungary
Received: 28 March 2011
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.
Mathematics Subject Classification: 62G05 / 62G99
Key words: Minkowski content / nonparametric set estimation / boundary estimation
The authors are grateful to Elena Villa for some interesting remarks and for pointing out some inaccuracies. The useful and constructive reports from two referees and an associate editor are also gratefully acknowledged. The work of A. Cuevas and R. Fraiman has been partially supported by Spanish grants MTM2010-17366 and CCG10-UAM/ESP-5494 (A. Cuevas). The work of L. Györfi was supported in part by the Computer and Automation Research Institute of the Hungarian Academy of Sciences and by the PASCAL2 Network of Excellence under EC Grant No. 216886.
© EDP Sciences, SMAI, 2013
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