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Central limit theorems for a class of symmetric statistics

Published online by Cambridge University Press:  24 October 2008

N. C. Weber
N. C. Weber
Affiliation:
Department of Mathematical Statistics, University of Sydney, Australia
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Abstract

Motivated by problems in the analysis of spatial data, central limit theorems are developed for U-statistics whose kernels depend on the size of the observed sample. These theorems are then applied to obtain results for the interpoint distance statistic and the large angle statistic.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 2 , September 1983 , pp. 307 - 313
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

REFERENCES

(1)Bowman, A. W. Some aspects of density estimation by the kernel method. Ph.D. thesis, University of Glasgow, 1981.Google Scholar
(2)Broadbent, S.Simulating the ley hunter. J. Boy. Statist. Soc. A 143 (1980), 109140.CrossRefGoogle Scholar
(3)Brown, T. C. and Silverman, B. W.Rates of Poisson convergence for U-statistics. J. Appl.Probab. 16 (1979), 428432.CrossRefGoogle Scholar
(4)Doob, J. L.Stochastic processes (Wiley, 1953).Google Scholar
(5)Funk, G. M. and Saunders, R. Central limit theorems for a clustering statistic. (1979.) (Preprint.)Google Scholar
(6)Hoeffding, W.A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 (1948), 293325.CrossRefGoogle Scholar
(7)Kendall, D. G. and Kendall, W. S.Alignments in two dimensional random sets of points. Adv. in Appl. Probab. 12 (1980), 380424.CrossRefGoogle Scholar
(8)Kester, A.Asymptotic normality of the number of small distances between random points in a cube. Stock. Proc. Appl. 3 (1975), 4554.CrossRefGoogle Scholar
(9)Santal, L.Integral geometry and geometric probability (Addison-Wesley, 1976).Google Scholar
(10)Scott, D. J.Central limit theorems for martingales 8nd for processes with stationary increments using a Skorokhod representation approach. Adv. in Appl. Probab. 5 (1973), 119137.CrossRefGoogle Scholar
(11)Silverman, B. W. and Brown, T. C.Short distances, flat triangles and Poisson limits. J. Appl. Probab. 15 (1978), 815825.CrossRefGoogle Scholar
(12)Weber, N. C.Rates of convergence for U-statistics with varying kernels. Bull. Austral. Math. Soc. 21 (1980), 15.CrossRefGoogle Scholar