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A boundary corrected expansion of the moments of nearest neighbor distributions
Author(s) -
Liitiäinen Elia,
Lendasse Amaury,
Corona Francesco
Publication year - 2010
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20311
Subject(s) - statistical physics , algebraic number , mathematics , k nearest neighbors algorithm , boundary (topology) , entropy (arrow of time) , principle of maximum entropy , moment (physics) , mathematical analysis , computer science , statistics , physics , classical mechanics , quantum mechanics , artificial intelligence
Abstract In this article, the moments of nearest neighbor distance distributions are examined. While the asymptotic form of such moments is well‐known, the boundary effect has this far resisted a rigorous analysis. Our goal is to develop a new technique that allows a closed‐form high order expansion, where the boundaries are taken into account up to the first order. The resulting theoretical predictions are tested via simulations and found to be much more accurate than the first order approximation obtained by neglecting the boundaries. While our results are of theoretical interest, they definitely also have important applications in statistics and physics. As a concrete example, we mention estimating Rényi entropies of probability distributions. Moreover, the algebraic technique developed may turn out to be useful in other, related problems including estimation of the Shannon differential entropy.© 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010