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Nonparametric estimation of boundary measures and related functionals: asymptotic results

Part of: Nonparametric inference Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Inés Armendáriz ,
Antonio Cuevas  and
Ricardo Fraiman
Inés Armendáriz*
Affiliation:
Universidad de San Andrés and Universidade de São Paulo
Antonio Cuevas*
Affiliation:
Universidad Autónoma de Madrid
Ricardo Fraiman*
Affiliation:
Universidad de San Andrés and Universidad de la República
*
∗ Postal address: Departamento de Matemática y Ciencias, Universidad de San Andrés, Vito Dumas 284 (B1644BID), Buenos Aires, Argentina.
∗∗∗ Postal address: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain. Email address: antonio.cuevas@uam.es
∗ Postal address: Departamento de Matemática y Ciencias, Universidad de San Andrés, Vito Dumas 284 (B1644BID), Buenos Aires, Argentina.
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Abstract

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We study a nonparametric method for estimating the boundary measure of a compact body G ⊂ ℝd (the boundary length when d = 2 and the surface area for d = 3) in the case when this measure agrees with the corresponding Minkowski content. The estimator we consider is closely related to the one proposed in Cuevas, Fraiman and Rodríguez-Casal (2007). Our method relies on two sets of random points, drawn inside and outside the set G, with different sampling intensities. Strong consistency and asymptotic normality are obtained under some shape hypotheses on the set G. Some applications and practical aspects are briefly discussed.

Keywords

Minkowski contentnonparametric set estimationstatistical image analysis

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 62G07: Density estimation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 311 - 322
Copyright
Copyright © Applied Probability Trust 2009 

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