A nonparametric approach to the estimation of lengths and surface areas

Details A nonparametric approach to the estimation of lengths and surface areas A nonparametric approach to the estimation of lengths and surface areas

2007-08-16

arxiv.org/abs/0708.2180v1

Annals of Statistics 2007, Vol. 35, No. 3, 1031-1051

Mathematics

Statistics Theory

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

Scientific paper

10.1214/009053606000001532

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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