On the estimation of a star-shaped set

Abstract

The estimation of a star-shaped set S from a random sample of points X 1,…,X n ∊ S is considered. We show that S can be consistently approximated (with respect to both the Hausdorff metric and the ‘distance in measure’ between sets) by an estimator ŝ n defined as a union of balls centered at the sample points with a common radius which can be chosen in such a way that ŝ n is also star-shaped. We also prove that, under some mild conditions, the topological boundary of the estimator ŝ n converges, in the Hausdorff sense, to that of S; this has a particular interest when the proposed estimation problem is considered from the point of view of statistical image analysis.