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Length and surface area estimation under smoothness restrictions

Part of: Nonparametric inference

Published online by Cambridge University Press:  01 July 2016

Beatriz Pateiro-López  and
Alberto Rodríguez-Casal
Beatriz Pateiro-López*
Affiliation:
Universidade de Santiago de Compostela
Alberto Rodríguez-Casal*
Affiliation:
Universidade de Santiago de Compostela
*
∗ Postal address: Departamento de Estatística e Investigación Operativa, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, 15782, Spain.
∗∗ Email address: alrodcas@usc.es
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Abstract

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The problem of estimating the Minkowski content L0(G) of a body G ⊂ ℝd is considered. For d = 2, the Minkowski content represents the boundary length of G. It is assumed that a ball of radius r can roll inside and outside the boundary of G. We use this shape restriction to propose a new estimator for L0(G). This estimator is based on the information provided by a random sample, taken on a square containing G, in which we know whether a sample point is in G or not. We obtain the almost sure convergence rate for the proposed estimator.

Keywords

Minkowski contentnonparametric set estimationlength estimationconvex setr-convexity

MSC classification

Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 2 , June 2008 , pp. 348 - 358
Copyright
Copyright © Applied Probability Trust 2008 

References

Baddeley, A. and Jensen, E. B. V. (2005). Stereology for Statisticians. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Bräker, H. and Hsing, T. (1998). On the area and perimeter of a random hull in a bounded convex set. Prob. Theory Relat. Fields 111, 517–550.CrossRefGoogle Scholar
Cuevas, A. and Rodríguez-Casal, A. (2003). Set estimation: an overview and some recent developments. In Recent Advances and Trends in Nonparametric Statistics, eds Akritas, M. G. and Politis, D. N., Elsevier, Amsterdam, pp. 251–264.Google Scholar
Cuevas, A., Fraiman, R. and Rodríguez-Casal, A. (2007). A nonparametric approach to the estimation of lengths and surface areas. Ann. Statist. 35, 1031–1051.Google Scholar
Dümbgen, L. and Walther, G. (1996). Rates of convergence for random approximations of convex sets. Adv. Appl. Prob. 28, 384–393.Google Scholar
Federer, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93, 418–491.Google Scholar
Mattila, P. (1995). Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press.Google Scholar
Rodríguez-Casal, A. (2007). Set estimation under convexity type assumptions. Ann. Inst. H. Poincaré Prob. Statist. 43, 763–774.Google Scholar
Schneider, R. (1988). Random approximation of convex sets. J. Microscopy 151, 211–227.Google Scholar
Serra, J. (1982). Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Walther, G. (1999). On a generalization of Blaschke's rolling theorem and the smoothing of surfaces. Math. Methods Appl. Sci. 22, 301–316.Google Scholar