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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.
@article{PS_2013__17__359_0,
author = {Cuevas, Antonio and Fraiman, Ricardo and Gy\"orfi, L\'aszl\'o},
title = {Towards a universally consistent estimator of the {Minkowski} content},
journal = {ESAIM: Probability and Statistics},
pages = {359--369},
publisher = {EDP-Sciences},
volume = {17},
year = {2013},
doi = {10.1051/ps/2011160},
mrnumber = {3066384},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2011160/}
}
TY - JOUR AU - Cuevas, Antonio AU - Fraiman, Ricardo AU - Györfi, László TI - Towards a universally consistent estimator of the Minkowski content JO - ESAIM: Probability and Statistics PY - 2013 SP - 359 EP - 369 VL - 17 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/ps/2011160/ DO - 10.1051/ps/2011160 LA - en ID - PS_2013__17__359_0 ER -
%0 Journal Article %A Cuevas, Antonio %A Fraiman, Ricardo %A Györfi, László %T Towards a universally consistent estimator of the Minkowski content %J ESAIM: Probability and Statistics %D 2013 %P 359-369 %V 17 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/ps/2011160/ %R 10.1051/ps/2011160 %G en %F PS_2013__17__359_0
Cuevas, Antonio; Fraiman, Ricardo; Györfi, László. Towards a universally consistent estimator of the Minkowski content. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 359-369. doi: 10.1051/ps/2011160
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