Towards a universally consistent estimator of the minkowski content

Abstract

We deal with a subject in the interplay between nonparametric statistics and geometric measure theory. The measure L0(G) of the boundary of a set G ⊂ ℝd (with d ≥ 2) can be formally defined, via a simple limit, by the so-called Minkowski content. We study the estimation of L0(G) from a sample of random points inside and outside G. The sample design assumes that, for each sample point, we know (without error) whether or not that point belongs to G. Under this design we suggest a simple nonparametric estimator and investigate its consistency properties. The main emphasis in this paper is on generality. So we are especially concerned with proving the consistency of our estimator under minimal assumptions on the set G. In particular, we establish a mild shape condition on G under which the proposed estimator is consistent in L2. Roughly speaking, such condition establishes that the set of gvery spikyh points at the boundary of G must be gsmallh. This is formalized in terms of the Minkowski content of such set. Several examples are discussed. © EDP Sciences, SMAI 2013.