A geometrically motivated parametric model in manifold estimation,

Abstract

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 $S$. We will assume that the sample data are given by the distances to the $(d-1)$-dimensional manifold $S$ from points randomly chosen on a band surrounding $S$, with $d=2$ and $d=3$. The point in this paper is to show that, if $S$ belongs to a wide class of compact sets (which we call \it sets with polynomial volume\rm), 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 behavior: while they are consistent and asymptotically normal, their expectations are infinite. The theoretical and practical consequences of this fact are discussed in some detail.