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Mutual and shared neighbor probabilities: finite- and infinite-dimensional results

Published online by Cambridge University Press:  01 July 2016

M. F. Schilling
M. F. Schilling*
Affiliation:
California State University, Northridge
*
∗Postal address: Department of Mathematics, School of Science and Mathematics, California State University, Northridge, 18111 Nordhoff St, Northridge, CA 91330, USA.
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Abstract

Let X1, ···, Xn be i.i.d. random variables defined in ℝd having common continuous density f(x), and let Rij be the rank of Xj in the ordered list of distances from X¡. Both the mutual neighbor probabilities p1(r, s) = P(R12 = r, R21 = s) and the neighbor-sharing probabilities p2(r, s) = P(R13 = r, R23 = s) are studied from an asymptotic viewpoint. Infinite-dimensional limits are found for both situations and take particularly simple forms. Both cases exhibit considerable stability across dimensions and thus are well approximated by their infinite-dimensional values. Tables are provided to support the results given.

Keywords

NEAREST NEIGHBORSGEOMETRIC PROBABILITY
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 2 , June 1986 , pp. 388 - 405
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

Research supported in part by National Science Foundation Grants MCS79-19141, MCS80-17103.

References

Clark, P. J. (1956) Grouping in spatial distributions. Science 123, 373–374.Google Scholar
Clark, P. J. and Evans, F. C. (1955) On some aspects of spatial pattern in biological populations. Science 121, 397–398.Google Scholar
Cox, T. F. (1976) The robust estimation of the density of a forest stand using a new conditioned distance method. Biometrika 63, 493–499.Google Scholar
Cox, T. F. (1981) Reflexive nearest neighbours. Biometrics 37, 367–369.Google Scholar
Cox, T. F. and Lewis, T. (1976) A conditioned distance ratio method for analyzing spatial patterns. Biometrika 73, 483–491.Google Scholar
Dacey, M. F. (1969) Proportion of reflexive nth order neighbours in a spatial distribution. Geographical Analysis 1, 385–388.Google Scholar
Diggle, P. J. (1975) Robust density estimation using distance methods. Biometrika 62, 39–48.CrossRefGoogle Scholar
Newman, C. M., Rinott, Y. and Tversky, A. (1983) Nearest neighbors and Voronoi regions in certain point processes. Adv. Appl. Prob. 15, 726–751.Google Scholar
Pickard, D. K. (1982) Isolated nearest neighbors. J. Appl. Prob. 19, 444–449.Google Scholar
Roberts, F. D. K. (1969) Nearest neighbours in a Poisson ensemble. Biometrika 56, 401–406.CrossRefGoogle Scholar
Schilling, M. F. (1983) An infinite-dimensional approximation for nearest neighbor goodness of fit tests. Ann. Statist. 11, 13–24.Google Scholar
Schilling, M. F. (1986) Multivariate two-sample tests based on nearest neighbors. J. Amer. Statist. Assoc. To appear.Google Scholar
Schwarz, G. and Tversky, A. (1980) On the reciprocity of proximity relations. J. Math. Psychol. 22, 157–175.Google Scholar