Surface area estimation under convexity type assumptions

B. Pateiro-López and A. Rodríguez-Casal

Journal of Nonparametric Statistics, 2009, vol. 21, issue 6, 729-741

Abstract: The problem of estimating the surface area, L0, of a set G⊂ℝd has been extensively considered in several fields of research. For example, stereology focuses on the estimation of L0 without needing to reconstruct the set G. From a more geometrical point of view, set estimation theory is interested in estimating the shape of the set. Thus, surface area estimation can be seen as a further step where the emphasis is placed on an important geometric characteristic of G. The Minkowski content is an attractive way to define L0 that has been previously used in the literature on surface area estimation. Pateiro-López and Rodríguez-Casal [B. Pateiro-López and A. Rodríguez-Casal, Length and surface area estimation under smoothness restrictions, Adv. Appl. Prob. 40(2) (2008), pp. 348–358] proposed an estimator, Ln, for L0 under convexity type assumptions. In this paper, we obtain the L1-convergence rate of Ln.

Date: 2009
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DOI: 10.1080/10485250902971732

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