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Some peculiar boundary phenomena for extremes of rth nearest neighbor links
Authors:H. Dette

N. Henze

Affiliation:

Abteilung Medizinische Statistik, Universität Göttingen, D-3400 Göttingen, FR Germany

Institut für Mathematische Stochastik, D-3000 Hannover 1, FR Germany

Abstract:Let Dn,r denote the largest rth nearest neighbor link for n points drawn independently and uniformly from the unit d-cube Cd. We show that according as r < d or r>d, the limiting behavior of Dn,r, as n → ∞, is determined by the two-dimensional ‘faces’ respectively one-dimensional ‘edges’ of the boundary of Cd. If d = r, a ‘balance’ between faces and edges occurs. In case of a d-dimensional sphere (instead of a cube) the boundary dominates the asymptotic behavior of Dn,r if d 3 or if d = 2, r 3.
Keywords:Computational geometry   nearest neighbor distances   extreme-value distribution   boundary domination   limit theorem
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