On boundary estimation

Abstract

We consider the problem of estimating the boundary of a compact set S ⊂ ℝ d from a random sample of points taken from S. We use the Devroye-Wise estimator which is a union of balls centred at the sample points with a common radius (the smoothing parameter in this problem). A universal consistency result, with respect to the Hausdorff metric, is proved and convergence rates are also obtained under broad intuitive conditions of a geometrical character. In particular, a shape condition on S, which we call expandability, plays an important role in our results. The simple structure of the considered estimator presents some practical advantages (for example, the computational identification of the boundary is very easy) and makes this problem quite close to some basic issues in stochastic geometry.