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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: [email protected]
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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 

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