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On the Distributions of Scan Statistics of a Two-Dimensional Poisson Process

Part of: Distribution theory Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Sven Erick Alm
Sven Erick Alm*
Affiliation:
Uppsala University
*
∗ Postal address: Department of Mathematics, Uppsala University, PO Box 480, S-751 06 Uppsala, Sweden.
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Abstract

Given a two-dimensional Poisson process, X, with intensity λ, we are interested in the largest number of points, L, contained in a translate of a fixed scanning set, C, restricted to lie inside a rectangular area.

The distribution of L is accurately approximated for rectangular scanning sets, using a technique that can be extended to higher dimensions. Reasonable approximations for non-rectangular scanning sets are also obtained using a simple correction of the rectangular result.

Keywords

POISSON PROCESSSCAN STATISTICAPPROXIMATION

MSC classification

Primary: 60G55: Point processes 60G70: Extreme value theory; extremal processes
Secondary: 62E17: Approximations to distributions (nonasymptotic)
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997 , pp. 1 - 18
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

Research supported by the Axel and Margaret Ax:son Johnson Foundation.

References

[1] Aldous, D. (1989) Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.Google Scholar
[2] Alm, S. E. (1983) On the distribution of the scan statistic of a Poisson process. In Probability and Mathematical Statistics: Essays in Honour of Carl-Gustav Esseen. ed. Gut, A. and Holst, L.. pp. 1–10.Google Scholar
[3] Huntington, R. J. and Naus, J. I. (1975) A simpler expression for kth nearest-neighbor coincidence probabilities. Ann. Prob. 3, 894–896.Google Scholar
[4] Janson, S. (1984) Bounds on the distributions of extremal values of a scanning process. Stoch. Proc. Appl. 18, 313–328.Google Scholar
[5] Loader, C. R. (1991) Large-deviation approximations to the distribution of scan statistics. Adv. Appl. Prob. 23, 751–771.Google Scholar
[6] Mack, C. (1949) The expected number of aggregates in a random distribution of n points. Proc. Cambridge Phil. Soc. 46, 285–292.Google Scholar
[7] Månsson, M. (1994) Covering uniformly distributed points by convex scanning sets. Preprint 1994:17/ISSN 0347-2809. Chalmers University of Technology.Google Scholar
[8] Månsson, M. (1995) Intersections of uniformly distributed translations of convex sets in two and three dimensions. Preprint 1995:11/ISSN 0347-2809. Chalmers University of Technology.Google Scholar
[9] Naus, J. I. (1982) Approximations for distributions of scan statistics. J. Amer. Statist. Assoc. 77, 177–183.Google Scholar