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A Poisson limit theorem for incomplete symmetric statistics

Published online by Cambridge University Press:  14 July 2016

M. Berman  and
G. K. Eagleson
M. Berman*
Affiliation:
CSIRO Division of Mathematics and Statistics, Sydney
G. K. Eagleson*
Affiliation:
CSIRO Division of Mathematics and Statistics, Sydney
*
∗ Postal address: CSIRO Division of Mathematics and Statistics, P.O. Box 218, Lindfield NSW 2070, Australia.
∗ Postal address: CSIRO Division of Mathematics and Statistics, P.O. Box 218, Lindfield NSW 2070, Australia.
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Abstract

Silverman and Brown (1978) have derived Poisson limit theorems for certain sequences of symmetric statistics, based on a sample of independent identically distributed random variables. In this paper an incomplete version of these statistics is considered and a Poisson limit result shown to hold. The powers of some tests based on the incomplete statistic are investigated and the main results of the paper are used to simplify the derivations of the asymptotic distributions of some statistics previously published in the literature.

Keywords

M-DEPENDENT PROCESSPOINT PROCESSPOISSON PROCESSRANDOM COVERINGRENEWAL PROCESSSCAN STATISTIC
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 1 , March 1983 , pp. 47 - 60
Copyright
Copyright © Applied Probability Trust 1983 

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References

Adler, R. J. (1978) Weak convergence results for extremal processes generated by dependent random variables. Ann. Prob. 6, 660–667.CrossRefGoogle Scholar
Aldous, D. J. (1978) Weak convergence of randomly indexed sequences of random variables. Math. Proc. Camb. Phil. Soc. 83, 117–126.Google Scholar
Blom, G. (1976) Some properties of incomplete U-statistics. Biometrika 63, 573–580.CrossRefGoogle Scholar
Brown, B. M. and Kildea, D. G. (1978) Reduced U-statistics and the Hodges–Lehmann estimator. Ann. Statist. 6, 828–835.Google Scholar
Cressie, N. (1977a) The minimum of higher order gaps. Austral. J. Statist. 19, 132–143.Google Scholar
Cressie, N. (1977b) On some properties of the scan statistic on the circle and the line. J. Appl. Prob. 14, 272–283.Google Scholar
Cressie, N. (1980) The asymptotic distribution of the scan statistic under uniformity. Ann. Prob. 8, 828–840.Google Scholar
Eagleson, G. K. and Weber, N. C. (1978) Limit theorems for weakly exchangeable arrays. Math. Proc. Camb. Phil. Soc. 84, 123–130.CrossRefGoogle Scholar
Flatto, L. (1973) A limit theorem for random coverings of a circle. Israel J. Math. 15, 167–184.Google Scholar
Galambos, J. (1978) The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.Google Scholar
Hoeffding, W. (1948) A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19, 293–325.Google Scholar
Huntington, R. J. and Naus, J. I. (1975) A simpler expression for kth nearest neighbour coincidence probabilities. Ann. Prob. 3, 894–896.Google Scholar
Jagers, P. and Lindvall, T. (1974) Thinning and rare events in point processes. Z. Wahrscheinlichskeitsth. 28, 89–98.Google Scholar
Janson, S. (1980) The asymptotic distribution of incomplete U-statistics. Preprint.Google Scholar
Kaplan, N. (1977) Two applications of a Poisson approximation for dependent events. Ann. Prob. 5, 787–794.Google Scholar
Lee, A. J. (1980) On the asymptotic distribution of certain incomplete U-statistics. Preprint.Google Scholar
Naus, J. I. (1965) The distribution of the size of the maximum cluster of points on a line. J. Amer. Statist. Assoc. 60, 532–538.Google Scholar
Naus, J. I. (1966a) Some probabilities, expectations and variances for the size of largest clusters and smallest intervals. J. Amer. Statist. Assoc. 61, 1191–1199.Google Scholar
Naus, J. I. (1966b) A power comparison of two tests of non random clustering. Technometrics 8, 493–517.Google Scholar
Newell, G. F. (1963) Distribution for the smallest distance between any pair of kth nearest-neighbour random points on a line. Proc. Symp. Time Series Analysis (Brown University, 1962), Wiley, New York, 89–103.Google Scholar
Ripley, B. D. (1977) Modelling spatial patterns. J. R. Statist. Soc. B 39, 179–212.Google Scholar
Ripley, B. D. and Silverman, B. W. (1978) Quick tests for spatial interaction. Biometrika 65, 641–642.Google Scholar
Silverman, B. W. and Brown, T. C. (1978) Short distances, flat triangles and Poisson limits. J. Appl. Prob. 15, 815–825.Google Scholar
Wallenstein, S. R. and Naus, J. I. (1974) Probabilities of the size of largest clusters and smallest intervals. J. Amer. Statist. Assoc. 69, 690–695.CrossRefGoogle Scholar
Weber, N. C. (1981) Incomplete degenerate U-statistics. Scand. J. Statist. 8, 120–123.Google Scholar