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Abstrakty
For a random sample of points in R, we consider the number, of pairs whose members are nearest neighbors (NNs) to each other and the, number of pairs sharing a common NN. The pairs of the first type are called, reflexive NNs, whereas the pairs of the latter type are called shared NNs. In, this article, we consider the case where the random sample of size n is from, the uniform distribution on an interval. We denote the number of reflexive NN pairs and the number of shared NN pairs in the sample by Rn and Qn, respectively. We derive the exact forms of the expected value and the variance for both Rn and Qn, and derive a recurrence relation for Rn which may also be used to compute the exact probability mass function (pmf) of Rn. Our approach is a novel method for finding the pmf of Rn and agrees with the results in the literature. We also present SLLN and CLT results for both Rn and Qn as n goes to infinity.
Czasopismo
Rocznik
Tom
Strony
123--137
Opis fizyczny
Bibliogr. 26 poz.
Twórcy
autor
- Department of Mathematics and Computer Science, Yıldırım Beyazıt University, Ankara, 06010, Turkey
autor
- Department of Statistics, University of Pittsburgh, Pittsburgh, PA, 15260, USA
Bibliografia
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- [3] E. Ceyhan, Overall and pairwise segregation tests based on nearest neighbor contingency tables, Comput. Statist. Data Anal. 53 (8) (2008), pp. 2786-2808.
- [4] E. Ceyhan, Testing spatial symmetry using contingency tables based on nearest neighbor relations, Sci. World J. (2014), Article ID 698296.
- [5] G. Chartrand and L. Lesniak, Graphs & Digraphs, Chapman & Hall/CRC, Boca Raton 1996.
- [6] K. L. Chung, A Course in Probability Theory, Academic Press, New York 1974.
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- [13] D. Eppstein, M. S. Paterson, and F. F. Yao, On nearest-neighbor graphs, Discrete Comput. Geom. 17 (3) (1997), pp. 263-282.
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- [16] W. Hoeffding, A class of statistics with asymptotically normal distribution, Ann. Math. Statist. 19 (3) (1948), pp. 293-325.
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- [18] C. Houdré and R. Restrepo, A probabilistic approach to the asymptotics of the length of the longest alternating subsequence, Electron. J. Combin. 17 (1) (2010), Research paper 168.
- [19] T. Kozakova, R. Meester, and S. Nanda, The size of components in continuum nearest-neighbor graphs, Ann. Probab. 34 (2) (2006), pp. 528-538.
- [20] C. M. Newman, Y. Rinott, and A. Tversky, Nearest neighbors and Voronoi regions in certain point processes, Adv. in Appl. Probab. 15 (4) (1983), pp. 726-751.
- [21] M. D. Penrose and J. E. Yukich, Central limit theorems for some graphs in computational geometry, Ann. Appl. Probab. 11 (4) (2001), pp. 1005-1041.
- [22] D. K. Pickard, Isolated nearest neighbors, J. Appl. Probab. 19 (2) (1982), pp. 444-449.
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Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-9a80cdf0-5b0a-4789-acf3-2dbed4f21eab