Towards a universally consistent estimator of the Minkowski content^∗

¹ Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain
antonio.cuevas@uam.es
² Departamento de Matemáticas y Ciencias, Universidad de San Andrés, Argentina and CMAT, Universidad de la República, Uruguay
³ Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Hungary

Received: 28 March 2011

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 ⊂ ℝ^d (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 “very spiky” points at the boundary of G must be “small”. This is formalized in terms of the Minkowski content of such set. Several examples are discussed.