Advances in Applied Probability
- Adv. in Appl. Probab.
- Volume 30, Number 2 (1998), 295-316.
On the Hausdorff distance between a convex set and an interior random convex hull
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
(Δn - 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.
Adv. in Appl. Probab. Volume 30, Number 2 (1998), 295-316.
First available in Project Euclid: 21 October 2002
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Bräker, H.; Hsing, T.; Bingham, N. H. On the Hausdorff distance between a convex set and an interior random convex hull. Adv. in Appl. Probab. 30 (1998), no. 2, 295--316. doi:10.1239/aap/1035228070. http://projecteuclid.org/euclid.aap/1035228070.