Towards a universally consistent estimator of the Minkowski content
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 359-369.

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.

DOI : 10.1051/ps/2011160
Classification : 62G05, 62G99
Mots-clés : Minkowski content, nonparametric set estimation, boundary estimation 
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     title = {Towards a universally consistent estimator of the {Minkowski} content},
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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. http://www.numdam.org/articles/10.1051/ps/2011160/

[1] L. Ambrosio, A. Colesanti and E. Villa, Outer Minkowski content for some classes of closed sets. Math. Ann. 342 (2008) 727-748. | MR | Zbl

[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. | MR | Zbl

[3] A.J. Baddeley and E.B. Vedel-Jensen, Stereology for Statisticians. Chapman & Hall, London (2005). | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[7] P. Erdős, Some remarks on the measurability of certain sets. Bull. Amer. Math. Soc. 51 (1945) 728-731. | MR | Zbl

[8] H. Federer, Curvature measures. Trans. Amer. Math. Soc. 93 (1959) 418-491. | MR | Zbl

[9] H. Federer, Geometric Measure Theory. Springer, New York (1969). | MR | Zbl

[10] R. Jiménez and J.E. Yukich, Nonparametric estimation of surface integrals. Ann. Stat. 39 (2011) 232-260. | MR | Zbl

[11] E. Mammen and A.B. Tsybakov, Asymptotical minimax recovery of sets with smooth boundaries. Ann. Stat. 23 (1995) 502-524. | MR | Zbl

[12] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and rectifiability. Cambridge University Press, Cambridge (1995). | MR | Zbl

[13] T.C. O'Neill, Geometric measure theory, in Encyclopedia of Mathematics, Supplement III. Kluwer Academic Publishers (2002).

[14] B. Pateiro-López and A. Rodríguez-Casal, Length and surface area estimation under convexity type restrictions. Adv. Appl. Probab. 40 (2008) 348-358.

[15] E. Villa, On the outer Minkowski content of sets. Ann. Mat. Pura Appl. 188 (2009) 619-630. | MR | Zbl

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