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Title: On the Hausdorff distance between a convex set and an interior random convex hull
Authors: Bräker, H.
Hsing, T. 
Bingham, N.H.
Keywords: Convex hull
Convex set
Extreme value theory
Gumbel distribution
Hausdorff metric
Home range
Limit law
Moving boundary
Smooth boundary
Issue Date: Jun-1998
Citation: Bräker, H.,Hsing, T.,Bingham, N.H. (1998-06). On the Hausdorff distance between a convex set and an interior random convex hull. Advances in Applied Probability 30 (2) : 295-316. ScholarBank@NUS Repository.
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 Knm 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.
Source Title: Advances in Applied Probability
ISSN: 00018678
Appears in Collections:Staff Publications

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