Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T19:49:18.216Z Has data issue: false hasContentIssue false
  • English
  • Français

Towards a universally consistent estimatorof 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. [email protected]
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
Get access

Abstract

We deal with a subject in the interplay between nonparametric statistics and geometricmeasure theory. The measure L0(G) of theboundary of a set G ⊂ ℝd (withd ≥ 2) can be formally defined, via a simple limit, bythe so-called Minkowski content. We study the estimation ofL0(G) from a sample of random points insideand outside G. The sample design assumes that, for each sample point, weknow (without error) whether or not that point belongs to G. Under thisdesign we suggest a simple nonparametric estimator and investigate its consistencyproperties. The main emphasis in this paper is on generality. So we are especiallyconcerned with proving the consistency of our estimator under minimal assumptions on theset G. In particular, we establish a mild shape condition onG under which the proposed estimator is consistent inL2. Roughly speaking, such condition establishes that theset of “very spiky” points at the boundary of G must be “small”. This isformalized in terms of the Minkowski content of such set. Several examples arediscussed.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ambrosio, L., Colesanti, A. and Villa, E., Outer Minkowski content for some classes of closed sets. Math. Ann. 342 (2008) 727–748. Google Scholar
Armendáriz, I., Cuevas, A. and Fraiman, R., Nonparametric estimation of boundary measures and related functionals: asymptotic results. Adv. Appl. Probab. 41 (2009) 311–322. Google Scholar
A.J. Baddeley and E.B. Vedel-Jensen, Stereology for Statisticians. Chapman & Hall, London (2005).
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.
Cuevas, A., Fraiman, R. and Rodríguez-Casal, A., A nonparametric approach to the estimation of lengths and surface areas. Ann. Stat. 35 (2007) 1031–1051. Google Scholar
Cuevas, A., Fraiman, R. and Pateiro-López, B., On statistical properties of sets fulfilling rolling-type conditions. Adv. Appl. Probab. 44 (2012) 311–329. Google Scholar
Erdős, P., Some remarks on the measurability of certain sets. Bull. Amer. Math. Soc. 51 (1945) 728–731. Google Scholar
Federer, H., Curvature measures. Trans. Amer. Math. Soc. 93 (1959) 418–491. Google Scholar
H. Federer, Geometric Measure Theory. Springer, New York (1969).
Jiménez, R. and Yukich, J.E., Nonparametric estimation of surface integrals. Ann. Stat. 39 (2011) 232–260. Google Scholar
Mammen, E. and Tsybakov, A.B., Asymptotical minimax recovery of sets with smooth boundaries. Ann. Stat. 23 (1995) 502–524. Google Scholar
P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and rectifiability. Cambridge University Press, Cambridge (1995).
T.C. O’Neill, Geometric measure theory, in Encyclopedia of Mathematics, Supplement III. Kluwer Academic Publishers (2002).
Pateiro-López, B. and Rodríguez-Casal, A., Length and surface area estimation under convexity type restrictions. Adv. Appl. Probab. 40 (2008) 348–358. Google Scholar
Villa, E., On the outer Minkowski content of sets. Ann. Mat. Pura Appl. 188 (2009) 619–630. Google Scholar