Towards a universally consistent estimator of the Minkowski content

Antonio Cuevas; Ricardo Fraiman; László Györfi

ESAIM: Probability and Statistics (2013)

  • Volume: 17, page 359-369
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Cuevas, Antonio, Fraiman, Ricardo, and Györfi, László. "Towards a universally consistent estimator of the Minkowski content." ESAIM: Probability and Statistics 17 (2013): 359-369. <http://eudml.org/doc/273620>.

@article{Cuevas2013,
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.},
author = {Cuevas, Antonio, Fraiman, Ricardo, Györfi, László},
journal = {ESAIM: Probability and Statistics},
keywords = {Minkowski content; nonparametric set estimation; boundary estimation},
language = {eng},
pages = {359-369},
publisher = {EDP-Sciences},
title = {Towards a universally consistent estimator of the Minkowski content},
url = {http://eudml.org/doc/273620},
volume = {17},
year = {2013},
}

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
PB - EDP-Sciences
VL - 17
SP - 359
EP - 369
AB - 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.
LA - eng
KW - Minkowski content; nonparametric set estimation; boundary estimation
UR - http://eudml.org/doc/273620
ER -

References

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  5. [5] A. Cuevas, R. Fraiman and A. Rodríguez-Casal, A nonparametric approach to the estimation of lengths and surface areas. Ann. Stat.35 (2007) 1031–1051. Zbl1124.62017
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  10. [10] R. Jiménez and J.E. Yukich, Nonparametric estimation of surface integrals. Ann. Stat.39 (2011) 232–260. Zbl1209.62059MR2797845
  11. [11] E. Mammen and A.B. Tsybakov, Asymptotical minimax recovery of sets with smooth boundaries. Ann. Stat.23 (1995) 502–524. Zbl0834.62038MR1332579
  12. [12] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and rectifiability. Cambridge University Press, Cambridge (1995). Zbl0911.28005MR1333890
  13. [13] T.C. O’Neill, Geometric measure theory, in Encyclopedia of Mathematics, Supplement III. Kluwer Academic Publishers (2002). 
  14. [14] B. Pateiro-López and A. Rodríguez-Casal, Length and surface area estimation under convexity type restrictions. Adv. Appl. Probab.40 (2008) 348–358. Zbl05306868
  15. [15] E. Villa, On the outer Minkowski content of sets. Ann. Mat. Pura Appl.188 (2009) 619–630. Zbl1173.28002MR2533959

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