Berrendero, José R.; Cholaquidis, Alejandro; Cuevas, Antonio; Fraiman, Ricardo A geometrically motivated parametric model in manifold estimation. (English) Zbl 1367.62173 Statistics 48, No. 5, 983-1004 (2014). Summary: The general aim of manifold estimation is reconstructing, by statistical methods, an \(m\)-dimensional compact manifold \(S\) on \(\mathbb{R}^d\) (with \(m\leq d\)) or estimating some relevant quantities related to the geometric properties of \(S\). Focussing on the cases \(d=2\) and \(d=3\), with \(m=d\) or \(m=d-1\), we will assume that the data are given by the distances to \(S\) from points randomly chosen on a band surrounding \(S\). The aim of this paper is to show that, if \(S\) belongs to a wide class of compact sets (which we call sets with polynomial volume), the proposed statistical model leads to a relatively simple parametric formulation. In this setup, standard methodologies (method of moments, maximum likelihood) can be used to estimate some interesting geometric parameters, including curvatures and Euler characteristic. We will particularly focus on the estimation of the (\(d-1\))-dimensional boundary measure (in Minkowski’s sense) of \(S\). It turns out, however, that the estimation problem is not straightforward since the standard estimators show a remarkably pathological behaviour: while they are consistent and asymptotically normal, their expectations are infinite. The theoretical and practical consequences of this fact are discussed in some detail. Cited in 6 Documents MSC: 62H12 Estimation in multivariate analysis 62F10 Point estimation 62H11 Directional data; spatial statistics 62H35 Image analysis in multivariate analysis Keywords:estimation of boundary length; estimation of curvature; distance to boundary; volume function; remote sensing PDFBibTeX XMLCite \textit{J. R. Berrendero} et al., Statistics 48, No. 5, 983--1004 (2014; Zbl 1367.62173) Full Text: DOI arXiv Link References: [1] Genovese CR, J Mach Learn Res. 13 pp 1263– (2012) [2] DOI: 10.1214/12-AOS994 · Zbl 1274.62237 [3] Dey TK, Curve and surface reconstruction (2007) [4] DOI: 10.1093/acprof:oso/9780199232574.003.0011 [5] DOI: 10.1239/aap/1086957575 · Zbl 1045.62019 [6] DOI: 10.1214/009053606000001532 · Zbl 1124.62017 [7] DOI: 10.1239/aap/1214950207 · Zbl 1416.62201 [8] DOI: 10.1239/aap/1246886612 · Zbl 1173.62016 [9] DOI: 10.1214/10-AOS837 · Zbl 1209.62059 [10] DOI: 10.1007/s00208-008-0254-z · Zbl 1152.28005 [11] DOI: 10.1007/s00209-003-0597-9 · Zbl 1059.53061 [12] DOI: 10.1007/s10231-008-0093-2 · Zbl 1173.28002 [13] DOI: 10.1214/aos/1069362379 · Zbl 0881.62067 [14] Steiner, J. Über parallele Flächen. Monatsbericht der Akademie der Wissenschaften zu Berlin; 1840. p. 114–118. [15] DOI: 10.1090/S0002-9947-1959-0110078-1 [16] Hatcher A, Algebraic topology (2002) [17] Stachó LL, Acta Sci Math. 38 pp 365– (1976) [18] DOI: 10.1007/s00208-003-0497-7 · Zbl 1048.52003 [19] Lehmann EL, Theory of point estimation, 2. ed. (1998) [20] Huber PJ, Robust statistics (1980) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.