A nonparametric approach to the estimation of lengths and surface areas

Antonio Cuevas; Ricardo Fraiman; Alberto Rodríguez-Casal

doi:10.1214/009053606000001532

paper

A nonparametric approach to the estimation of lengths and surface areas

arXiv:0708.2180 · doi:10.1214/009053606000001532

Abstract

The Minkowski content $L_{0} (G)$ of a body $G \subset R^{d}$ 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.

Published at http://dx.doi.org/10.1214/009053606000001532 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)