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arXiv · 0708.2180

A nonparametric approach to the estimation of lengths and surface areas

Antonio Cuevas·Ricardo Fraiman·Alberto Rodríguez-Casal
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Abstract

The Minkowski content $L_0(G)$ of a body $G\subset{\mathbb{R}}^d$ represents the boundary length (for $d=2$) or the surface area (for $d=3$) of $G$. A method for estimating $L_0(G)$ 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, $L_1$-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. 2007-08-16. A nonparametric approach to the estimation of lengths and surface areas. https://doi.org/10.1214/009053606000001532

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