On the Hausdorff distance between a convex set and an interior random convex hull

Abstract

The problem of estimating an unknown compact convex set K in the plane, from a sample (X 1,···,X n ) of points independently and uniformly distributed over K, is considered. Let K n be the convex hull of the sample, Δ be the Hausdorff distance, and Δ n := Δ (K, K n ). Under mild conditions, limit laws for Δ n are obtained. We find sequences (a n ), (b n ) such that (Δ n - b n )/a n → Λ (n → ∞), where Λ is the Gumbel (double-exponential) law from extreme-value theory. As expected, the directions of maximum curvature play a decisive role. Our results apply, for instance, to discs and to the interiors of ellipses, although for eccentricity e < 1 the first case cannot be obtained from the second by continuity. The polygonal case is also considered.

On the area and perimeter of a random convex hull in a bounded convex set

“…The two can be handled in the same manner and so we focus on g nY1 . Make the observation that if is small enough, then there exist 1 …”

“…Hsing (1994) and Cabo and Groeneboom (1994), respectively, derived the asymptotic distributions of the area of u n for the disk and the polygonal cases. BraÈ ker, Hsing and Bingham (1995) considered the asymptotic distribution of the Hausdor distance between u and u n for both the smooth and the polygonal cases.…”

Kernel estimation of density level sets

“…One may find in the literature many types of estimators, such as the plug-in estimators [1,2,9,19,20], the estimators defined by an excess mass approach [18,[21][22][23]25,28], the "naive" estimators (for e.g. [11,12] or [29] for a more sophisticated version of this idea), or the estimators constructed using a convex hull of the sample [5,13]. However, most of these techniques have several disadvantages when evaluated against the two main criteria, namely statistical performance and computational feasibility.…”

On boundary estimation

“…, X n }, which, of course, only makes sense when S is assumed to be convex; see e.g. Dümbgen and Walther (1996), Bräker et al (1998) and references therein. If the strong assumption of convexity is not acceptable, we would need a more flexible general procedure.…”