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Biometrika 1969 56(2):401-406; doi:10.1093/biomet/56.2.401
© 1969 by Biometrika Trust
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Nearest neighbours in a Poisson ensemble

F. D. K. ROBERTS*

University of Liverpool
*Now at University of Victoria B.C.

Every point of a Poisson ensemble has a unique nearest neighbour. However, every point is not necessarily the nearest neighbour of precisely one other point. The problem considered here is that of determining the probability pn that a point of a Poisson ensemble is the nearest neighbour of precisely n other points. The problem is discussed for Poisson ensembles in 1, 2 and 3 dimensions. Except in the one-dimensional case1pn has not been found analytically. Some bounds on pn are provided and estimates of pn are obtained by a Monte Carlotechnique.


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