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Laws of Large Numbers and Nearest Neighbor Distances

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Advances in Directional and Linear Statistics

Abstract

We consider the sum of power weighted nearest neighbor distances in a sample of size n from a multivariate density f of possibly unbounded support. We give various criteria guaranteeing that this sum satisfies a law of large numbers for large n, correcting some inaccuracies in the literature on the way. Motivation comes partly from the problem of consistent estimation of certain entropies of f.

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Acknowledgements

Research of Matthew Penrose supported in part by the Alexander von Humboldt Foundation through a Friedrich Wilhelm Bessel Research Award. Research of J.E. Yukich supported in part by NSF grant DMS-0805570.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Mathew D. Penrose

Authors
  1. Mathew D. Penrose
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  2. J. E. Yukich
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Corresponding author

Correspondence to Mathew D. Penrose .

Editor information

Editors and Affiliations

  1. , Statistical Science, Cornell University, Comstock Hall 1190, Ithaca, NY 14853, 14853, USA

    Martin T. Wells

  2. , Applied Statistics Unit, Indian Statistical Institute, Barrackpore Trunk Road 203, Kolkata, 700035, India

    Ashis SenGupta

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Penrose, M.D., Yukich, J.E. (2011). Laws of Large Numbers and Nearest Neighbor Distances. In: Wells, M., SenGupta, A. (eds) Advances in Directional and Linear Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2628-9_13

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