A nonparametric approach to the estimation of lengths and surface areas

Antonio Cuevas, Ricardo Fraiman, Alberto Rodríguez-Casal

The Annals of Statistics · 2007 · Volume 35 · Issue 3

Abstract

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.

Keywords

62G07, 62G20, length estimation, Minkowski content, nonparametric set estimation, statistical image analysis

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