Length and surface area estimation under smoothness restrictions

Abstract

The problem of estimating the Minkowski content L 0 (G) of a body G ⊂ R d is considered. For d = 2, the Minkowski content represents the boundary length of G. It is assumed that a ball of radius r can roll inside and outside the boundary of G. We use this shape restriction to propose a new estimator for L 0 (G). This estimator is based on the information provided by a random sample, taken on a square containing G, in which we know whether a sample point is in G or not. We obtain the almost sure convergence rate for the proposed estimator.

Surface area estimation under convexity type assumptions

“…If the r-convexity restriction is imposed, a more sophisticated surface area estimator L n (also based on the Minkowski content) can be considered. The precise definition of L n and the result about the almost sure convergence rate of the estimator can be found in [6]. In this paper we obtain the L 1 -convergence rate of L n .…”

“…Additional geometric information on the set of interest may be also helpful for defining efficient estimators. This is the case of the estimator, L n , proposed in [6], which is defined under convexity type assumptions. In this work, we analyse the L 1 -convergence rate of L n .…”

Nonparametric estimation of boundary measures and related functionals: asymptotic results

“…As is quite standard in nonparametric estimation, our techniques require the use of a smoothing parameter. This paper is a further development of the ideas proposed in [8]; see also [6] and [14] for related approaches. Cuevas and Rodríguez-Casal [7] (see also the references therein) studied the problem of boundary approximation from a nonparametric perspective as well, but they did not provide results on the estimation of boundary measures.…”

“…The proof of the first statement in Theorem 1 follows from (8) together with Lemmas 1 and 2 and (14).…”

“…This coincides with the best rate found in [8] for the estimation of L 0 (G) under M1. A faster convergence rate is obtained in [14], assuming additional shape conditions on G which are incorporated into the estimator. Our procedure yields rates that come arbitrarily close to n 1/2d = m 1/3 ; see the suggested values for the parameters m and n in (4).…”