We study the problem of estimating the surface area of the boundary of a sufficiently smooth set when the available information is only a finite subset . 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 -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 , which is denoted by . More precisely, it is the -dimensional surface area of , as denoted by , which is proven to converge to the -dimensional surface area of . Moreover, can be computed using Crofton’s formula.
Our results depend on the Hausdorff distance between S and for the Devroye–Wise estimator, and the Hausdorff distance between and for the second estimator.
This research has been partially supported by grant FCE-1-2019-1-156054 (ANII-Uruguay).
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.
"Estimation of surface area." Electron. J. Statist. 16 (2) 3751 - 3788, 2022. https://doi.org/10.1214/22-EJS2031