Open Access
July 2007 A nonparametric approach to the estimation of lengths and surface areas
Antonio Cuevas, Ricardo Fraiman, Alberto Rodríguez-Casal
Ann. Statist. 35(3): 1031-1051 (July 2007). DOI: 10.1214/009053606000001532


The Minkowski content L0(G) of a body G⊂ℝd represents the boundary length (for d=2) or the surface area (for d=3) of G. A method for estimating L0(G) is proposed. It relies on a nonparametric estimator based on the information provided by a random sample (taken on a rectangle containing G) in which we are able to identify whether every point is inside or outside G. Some theoretical properties concerning strong consistency, L1-error and convergence rates are obtained. A practical application to a problem of image analysis in cardiology is discussed in some detail. A brief simulation study is provided.


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Antonio Cuevas. Ricardo Fraiman. Alberto Rodríguez-Casal. "A nonparametric approach to the estimation of lengths and surface areas." Ann. Statist. 35 (3) 1031 - 1051, July 2007.


Published: July 2007
First available in Project Euclid: 24 July 2007

zbMATH: 1124.62017
MathSciNet: MR2341697
Digital Object Identifier: 10.1214/009053606000001532

Primary: 62G07
Secondary: 62G20

Keywords: length estimation , Minkowski content , nonparametric set estimation , statistical image analysis

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 3 • July 2007
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