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Some peculiar boundary phenomena for extremes of rth nearest neighbor links

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  • Dette, H.
  • Henze, N.

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, the limiting behavior of Dn,r, as n --> [infinity], 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 [greater-or-equal, slanted] 3 or if d = 2, r [greater-or-equal, slanted] 3.

Suggested Citation

  • Dette, H. & Henze, N., 1990. "Some peculiar boundary phenomena for extremes of rth nearest neighbor links," Statistics & Probability Letters, Elsevier, vol. 10(5), pages 381-390, October.
    • Handle: RePEc:eee:stapro:v:10:y:1990:i:5:p:381-390
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    Cited by:

    1. Ebner, Bruno & Henze, Norbert & Yukich, Joseph E., 2018. "Multivariate goodness-of-fit on flat and curved spaces via nearest neighbor distances," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 231-242.

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