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2022 Estimation of surface area
Catherine Aaron, Alejandro Cholaquidis, Ricardo Fraiman
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Electron. J. Statist. 16(2): 3751-3788 (2022). DOI: 10.1214/22-EJS2031

Abstract

We study the problem of estimating the surface area of the boundary S of a sufficiently smooth set SRd when the available information is only a finite subset XnS. We propose two estimators. The first makes use of the Devroye–Wise support estimator and is based on Crofton’s formula, which, roughly speaking, states that the (d1)-dimensional surface area of a smooth enough set is the mean number of intersections of randomly chosen lines. For that purpose, we propose an estimator of the number of intersections of such lines with support based on the Devroye–Wise support estimators. The second surface area estimator makes use of the α-convex hull of Xn, which is denoted by Cα(Xn). More precisely, it is the (d1)-dimensional surface area of Cα(Xn), as denoted by |Cα(Xn)|d1, which is proven to converge to the (d1)-dimensional surface area of S. Moreover, |Cα(Xn)|d1 can be computed using Crofton’s formula.

Our results depend on the Hausdorff distance between S and Xn for the Devroye–Wise estimator, and the Hausdorff distance between S and Cα(Xn) for the second estimator.

Funding Statement

This research has been partially supported by grant FCE-1-2019-1-156054 (ANII-Uruguay).

Acknowledgments

We would like to thank a referee and the AE for careful proofreading, and their helpful and positive suggestions in a previous version of this manuscript.

Citation

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Catherine Aaron. Alejandro Cholaquidis. Ricardo Fraiman. "Estimation of surface area." Electron. J. Statist. 16 (2) 3751 - 3788, 2022. https://doi.org/10.1214/22-EJS2031

Information

Received: 1 October 2021; Published: 2022
First available in Project Euclid: 6 July 2022

Digital Object Identifier: 10.1214/22-EJS2031

Keywords: Crofton’s formula , Devroye–Wise estimator , Surface estimation , α-convex hull

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