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

Part of: General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

H. Bräker ,
T. Hsing  and
N. H. Bingham
H. Bräker*
Affiliation:
University of Bern
T. Hsing*
Affiliation:
Singapore National University
N. H. Bingham*
Affiliation:
Birkbeck College (Univ. London)
*
∗ Postal address: University of Bern, 3012 Bern, Switzerland.
∗∗ Postal address: Singapore National University, 10 Kent Ridge Crescent, Singapore 119260K.
∗∗∗ Birkbeck College (Univ. London), London WC1E 7HX, UK. Email address: n.bingham@statistics.bbk.ac.uk
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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

(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.

Keywords

Convex setconvex hullHausdorff metriclimit lawGumbel distributionextreme value theorysmooth boundarypolygonmoving boundaryhome range

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 30 , Issue 2 , June 1998 , pp. 295 - 316
Copyright
Copyright © Applied Probability Trust 1998 

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References

Bárány, I., (1989). Intrinsic volumes and f-vectors of random polytopes. Mathematische Annalen 285, 671–699.Google Scholar
Bárány, I. and Larman, D. G. (1988). Convex bodies, economic cap coverings and random polytopes. Mathematika 35, 274–291.Google Scholar
Bräker, H. and Hsing, T. (1995). Limit theorems for random convex hulls. Preprint.Google Scholar
Cabo, A. J. and Groeneboom, P. (1994). Limit theorems for functionals of convex hulls. Prob. Theory Rel. Fields 100, 31–55.Google Scholar
Carnal, H. and Hüsler, J. (1991). On the convex hull of n random points on the circle. J. Appl. Prob. 28, 231–237.Google Scholar
Dekkers, A. L. and de Haan, L. (1989). On the estimation of the extreme-value index and large quantile estimation. Ann. Statist. 17, 1795–1832.Google Scholar
Dümbgen, L. and Walther, G. (1996). Rates of convergence for random approximations of convex sets. Adv. Appl. Prob. 28, 384–393.Google Scholar
Einmahl, J. H. J., de Hahn, L. and Huang, X. (1993). Estimating a multidimensional extreme-value distribution. J. Multivariate Anal. 47, 173–195.CrossRefGoogle Scholar
Groeneboom, P. (1988). Limit theorems for convex hulls. Prob. Theory Rel. Fields 79, 329–368.Google Scholar
Gruber, P. M. (1983). In most cases approximation is irregular. Rend. Sem. Mat. Torino 41, 19–33.Google Scholar
Hsing, T. (1994). On the asymptotic distribution of the area outside a random convex hull in a disk. Ann. Appl. Prob. 4, 478–493.CrossRefGoogle Scholar
Kreyszig, E. (1959). Differential Geometry. University of Toronto Press.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1982). Extremal and Related Properties of Stationary Processes. Springer.Google Scholar
MacDonald, D. W., Ball, F. G. and Hough, N. G. (1980). The evaluation of home range size and configuration using radio tracking. In Handbook on Biotelemetry and Radio Tracking, eds. MacDonald, D. W. and Amlaner, C. J.. Pergamon Press, Oxford.Google Scholar
Moore, M. (1984). On the estimation of a convex set. Ann. Statist. 12, 1090–1099.CrossRefGoogle Scholar
O'Brien, G. L. (1987). Extreme values for stationary and Markov sequences. Ann. Prob. 15, 281–289.Google Scholar
Rényi, A. and Sulanke, R. (1963). Über die konvexe Hülle von n zufällig gewählten Punkten, I. Z. Wahrsch. verw. Gebiete 2, 75–84. (Reprinted in Selected Papers of Alfréd Rényi. (1976). Volume 3, 143–152. Budapest, Akadémiai Kiadó.).Google Scholar
Rényi, A. and Sulanke, R. (1964). Über die konvexe Hülle von n zufällig gewählten Punkten, II. Z. Wahrsch. verw. Gebiete 3, 138–147. (Reprinted in Selected Papers of Alfréd Rényi. (1976). Volume 3, 242–251. Budapest, Akadémiai Kiadó.).Google Scholar
Ripley, B. D. and Rasson, J. P. (1977). Finding the edge of a Poisson forest. J. Appl. Prob. 14, 483–491.CrossRefGoogle Scholar
Rootzén, H., (1988). Maxima and exceedances of stationary Markov chains. Adv. Appl. Prob. 20, 371–390.Google Scholar
Schneider, R. and Wieacker, J. A. (1980). Random polytopes in a convex body. Z. Wahrsch. verw. Gebiete 52, 69–73.Google Scholar
Smith, R. L. (1987). Estimating tails of probability distributions. Ann. Statist. 15, 1174–1207.Google Scholar
Smith, R. L., Tawn, J. A. and Yuen, H. K. (1990). Statistics of multivariate extremes. Int. Statist. Rev. 58, 47–58.Google Scholar