# Towards a universally consistent estimator of the Minkowski content

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

top

## Abstract

top
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

top

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

top
1. [1] L. Ambrosio, A. Colesanti and E. Villa, Outer Minkowski content for some classes of closed sets. Math. Ann.342 (2008) 727–748. Zbl1152.28005MR2443761
2. [2] I. Armendáriz, A. Cuevas and R. Fraiman, Nonparametric estimation of boundary measures and related functionals: asymptotic results. Adv. Appl. Probab.41 (2009) 311–322. Zbl1173.62016MR2541178
3. [3] A.J. Baddeley and E.B. Vedel-Jensen, Stereology for Statisticians. Chapman & Hall, London (2005). Zbl1086.62108MR2107000
4. [4] A. Cuevas, and R. Fraiman, Set estimation, in New Perspectives on Stochastic Geometry, edited by W.S. Kendall and I. Molchanov. Oxford University Press (2010) 374–397. Zbl1192.62164MR2654684
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
6. [6] A. Cuevas, R. Fraiman and B. Pateiro-López, On statistical properties of sets fulfilling rolling-type conditions. Adv. Appl. Probab.44 (2012) 311–329. Zbl1252.47089MR2977397
7. [7] P. Erdős, Some remarks on the measurability of certain sets. Bull. Amer. Math. Soc.51 (1945) 728–731. Zbl0063.01269MR13776
8. [8] H. Federer, Curvature measures. Trans. Amer. Math. Soc.93 (1959) 418–491. Zbl0089.38402MR110078
9. [9] H. Federer, Geometric Measure Theory. Springer, New York (1969). Zbl0874.49001MR257325
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

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.