Advances in Applied Probability

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

H. Bräker, T. Hsing, and N. H. Bingham

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The problem of estimating an unknown compact convex set K in the plane, from a sample (X1,···,Xn) of points independently and uniformly distributed over K, is considered. Let Kn be the convex hull of the sample, Δ be the Hausdorff distance, and Δn := Δ (K, Kn). Under mild conditions, limit laws for Δn are obtained. We find sequences (an), (bn) such that

n - bn)/an → 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.

Article information

Source
Adv. in Appl. Probab. Volume 30, Number 2 (1998), 295-316.

Dates
First available: 21 October 2002

Permanent link to this document
http://projecteuclid.org/euclid.aap/1035228070

Digital Object Identifier
doi:10.1239/aap/1035228070

Mathematical Reviews number (MathSciNet)
MR1642840

Zentralblatt MATH identifier
0912.60022

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05]

Keywords
Convex set convex hull Hausdorff metric limit law Gumbel distribution extreme value theory smooth boundary polygon moving boundary home range

Citation

Bräker, H.; Hsing, T.; Bingham, N. H. On the Hausdorff distance between a convex set and an interior random convex hull. Advances in Applied Probability 30 (1998), no. 2, 295--316. doi:10.1239/aap/1035228070. http://projecteuclid.org/euclid.aap/1035228070.


Export citation