On the Hausdorff distance between a convex set and an interior random convex hull
1998 ◽
Vol 30
(2)
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pp. 295-316
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Keyword(s):
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.
1998 ◽
Vol 30
(02)
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pp. 295-316
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Keyword(s):
1996 ◽
Vol 28
(02)
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pp. 384-393
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2001 ◽
Vol 70
(3)
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pp. 323-336
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1984 ◽
Vol 16
(02)
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pp. 324-346
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1974 ◽
Vol 6
(03)
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pp. 563-579
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1984 ◽
Vol 27
(2)
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pp. 233-237
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1988 ◽
Vol 37
(2)
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pp. 177-200
1985 ◽
Vol 17
(02)
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pp. 308-329
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