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Poisson approximations for the distribution and moments of ordered m-spacings

Part of: Distribution theory - Probability Nonparametric inference Geometric probability and stochastic geometry Distribution theory

Published online by Cambridge University Press:  14 July 2016

Joseph Glaz ,
Joseph Naus ,
Malgorzata Roos  and
Sylvan Wallenstein
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Abstract

This article investigates the accuracy of approximations for the distribution of ordered m-spacings for i.i.d. uniform observations in the interval (0, 1). Several Poisson approximations and a compound Poisson approximation are studied. The result of a simulation study is included to assess the accuracy of these approximations. A numerical procedure for evaluating the moments of the ordered m-spacings is developed and evaluated for the most accurate approximation.

Keywords

COMPOUND POISSON APPROXIMATIONSMOMENTS

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60E05: Distributions 62E20: Asymptotic distribution theory 62G30: Order statistics; empirical distribution functions
Type
Part 5 Statistical Studies
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 271 - 281
Copyright
Copyright © Applied Probability Trust 1994 

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References

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Applied Mathematics Series No. 55, National Bureau of Census, Washington, DC.Google Scholar
Aldous, D. (1989) Probability Approximations via the Poisson Clumping Heuristic. Springer-Verlag, New York.CrossRefGoogle Scholar
Arratia, R., Goldstein, L. and Gordon, L. (1989) Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Prob. 17, 9–25.CrossRefGoogle Scholar
Arratia, R., Goldstein, L. and Gordon, L. (1990) Poisson approximation and the Chen-Stein method. Statist. Sci. 5, 403–434.Google Scholar
Barbour, A. D., Chen, L. H. Y. and Loh, W. L. (1992a) Compound Poisson approximation for nonnegative random variables via Stein's method. Ann. Prob. 20, 1843–1866.CrossRefGoogle Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992b) Poisson Approximation. Oxford University Press.CrossRefGoogle Scholar
Barton, D. E. and David, F. N. (1956) Some notes on ordered random intervals. J. R. Statist. Soc. A18, 79–94.Google Scholar
Berman, M. and Eagleson, G. K. (1983) A Poisson limit theorem for incomplete symmetric statistics. J. Appl. Prob. 20, 47–60.CrossRefGoogle Scholar
Berman, M. and Eagleson, G. K. (1985) A useful upper bound for the tail probabilities when the sample size is large. J. Amer. Statist. Assoc. 80, 886–889.CrossRefGoogle Scholar
Chen, L. H. Y. (1975) Poisson approximations for dependent trials. Ann. Prob. 3, 534–545.CrossRefGoogle Scholar
Cressie, N. (1977a) The minimum of higher order gaps. Austral. J. Statist. 19, 132–143.CrossRefGoogle Scholar
Cressie, N. (1977b) On some properties of the scan statistic on the circle and the line. J. Appl. Prob. 14, 272–283.CrossRefGoogle Scholar
Cressie, N. (1980) The asymptotic distribution of the scan under uniformity. Ann. Prob. 8, 828–840.CrossRefGoogle Scholar
Cressie, N. (1984) Using the scan statistic to test uniformity. Colloq. Math. Soc. Janos Bolyai 45, Goodness of Fit, Debrecen, Hungary, pp. 87–100.Google Scholar
David, H. A. (1981) Order Statistics , 2nd edn. Wiley, New York.Google Scholar
Darling, D. A. (1953) On a class of problems related to the random division of an interval. Ann. Math. Statist. 24, 239–253.CrossRefGoogle Scholar
Dembo, A. and Karlin, S. (1992) Poisson approximations for r-scan processes. Ann. Appl. Prob. 2, 329–357.CrossRefGoogle Scholar
Frosini, B. V. (1981) Distribution of the smallest interval that contains a given cluster of points. Statistica 41, 255–280.Google Scholar
Gates, D. J. and Westcott, M. (1984) On the distribution of the scan statistics. J. Amer. Statist. Assoc. 79, 423–429.CrossRefGoogle Scholar
Glaz, J. (1989) Approximations and bounds for the distribution of the scan statistic. J. Amer. Statist. Assoc. 84, 560–566.CrossRefGoogle Scholar
Glaz, J. (1992a) Approximations for tail probabilities and moments of the scan statistic. Comput. Stat. Data Anal. 14, 213–227.CrossRefGoogle Scholar
Glaz, J. (1992b) Approximations for the tail probabilities of scan statistics. Technical report, University of Connecticut.CrossRefGoogle Scholar
Glaz, J. (1993) Extreme order statistics for a sequence of dependent random variables. Stochastic Inequalities , IMS Lecture Notes - Monograph Series, Hayward, CA. (in press).CrossRefGoogle Scholar
Glaz, J. and Naus, J. (1983) Multiple clusters on the line. Commun. Statist. - Theory Meth. 12, 1961–1986.CrossRefGoogle Scholar
Huntington, R. J. and Naus, J. I. (1975) A simpler expression for kth nearest neighbor coincidence probabilities. Ann. Prob. 3, 894–896.CrossRefGoogle Scholar
Krauth, J. (1988) An improved upper bound for the tail probability of the scan statistic for testing non-random clustering. Classification and Related Methods of Data Analysis. Proc. 1st Conf. Intern. Feder. Classif. Soc., Technical University of Aachen, Germany, 237–244.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Mcclure, D. E. (1976) Extreme non uniform spacings. Report No. 44 in Pattern Analysis, Brown University.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.CrossRefGoogle 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.CrossRefGoogle Scholar
Naus, J. I. (1966b) A power comparison of two tests of non-random clustering. Technometrics 8, 493–517.Google Scholar
Naus, J. I. (1982) Approximations for distributions of scan statistics. J. Amer. Statist. Assoc. 77, 177–183.CrossRefGoogle Scholar
Neff, N. D. and Naus, J. I. (1980) The distribution of the size of the maximum cluster of points on a line. Selected Tables in Mathematical Statistics , Vol. 6, AMS, Providence, RI.Google Scholar
Newell, G. F. (1963) Distribution for the smallest distance between any pair of kth nearestneighbor random points on a line. Proc. Symp. on Time Series , Brown University, pp. 89–103. Wiley, New York.Google Scholar
Pyke, R. (1965) Spacings. J. R. Statist. Soc. B27, 395–449.Google Scholar
Pyke, R. (1972) Spacings revisited. Proc. 6th Berkeley Symp. Math. Statist. Prob. 1, 417–427.Google Scholar
Reiss, R. D. (1989) Approximate Distributions of Order Statistics. Springer-Verlag, New York.CrossRefGoogle Scholar
Roos, M. (1993) Compound Poisson approximations for the numbers of extreme spacings. Adv. Appl. Prob. 25, 847–874.CrossRefGoogle Scholar
Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar
Wallenstein, S. R. and Naus, J. I. (1974) Probabilities for the size of largest clusters and smallest intervals. J. Amer. Statist. Assoc. 69, 690–697.CrossRefGoogle Scholar
Wallenstein, S. and Neff, N. (1987) An approximation for the distribution of the scan statistic. Statist. Medicine 6, 197–207.CrossRefGoogle ScholarPubMed
Weiss, L. (1959) The limiting joint distribution of the largest and the smallest sample spacings. Ann. Math. Statist. 30, 590–593.CrossRefGoogle Scholar