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 (X 1,···,X n ) of points independently and uniformly distributed over K, is considered. Let K n be the convex hull of the sample, Δ be the Hausdorff distance, and Δ n := Δ (K, K n ). Under mild conditions, limit laws for Δ n are obtained. We find sequences (a n ), (b n ) such that (Δ n - b n )/a n → 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.

[1]  Holger Rootzén,et al.  Maxima and exceedances of stationary Markov chains , 1988, Advances in Applied Probability.

[2]  A. Dekkers,et al.  Optimal choice of sample fraction in extreme-value estimation , 1993 .

[3]  Imre Bárány,et al.  Intrinsic volumes andf-vectors of random polytopes , 1989 .

[4]  Piet Groeneboom,et al.  Limit theorems for functionals of convex hulls , 1994 .

[5]  L. Dümbgen,et al.  RATES OF CONVERGENCE FOR RANDOM APPROXIMATIONS OF CONVEX SETS , 1996 .

[6]  Richard L. Smith Estimating tails of probability distributions , 1987 .

[7]  Piet Groeneboom,et al.  Limit theorems for convex hulls , 1988 .

[8]  Laurens de Haan,et al.  Estimating a multidimensional extreme-value distribution , 1993 .

[9]  B. Ripley,et al.  Finding the edge of a Poisson forest , 1977, Journal of Applied Probability.

[10]  L. Haan,et al.  On the Estimation of the Extreme-Value Index and Large Quantile Estimation , 1989 .

[11]  Marc Moore,et al.  On the Estimation of a Convex Set , 1984 .

[12]  Imre Bárány,et al.  CONVEX-BODIES, ECONOMIC CAP COVERINGS, RANDOM POLYTOPES , 1988 .

[13]  George L. O'Brien,et al.  Extreme Values for Stationary and Markov Sequences , 1987 .

[14]  ON THE CONVEX HULL OF n RANDOM POINTS ON A CIRCLE , 1991 .

[15]  Jonathan A. Tawn,et al.  Statistics of Multivariate Extremes , 1990 .

[16]  Tailen Hsing On the Asymptotic Distribution of the Area Outside a Random Convex Hull in a Disk , 1994 .

[17]  Rolf Schneider,et al.  Random polytopes in a convex body , 1980 .