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Large-deviation approximations to the distribution of scan statistics

Published online by Cambridge University Press:  01 July 2016

Clive R. Loader
Clive R. Loader*
Affiliation:
AT&T Bell Laboratories
*
∗Postal address: AT&T Bell Laboratories, Room 2C-279, 600 Mountain Avenue, Murray Hill, NJ 07974, USA.
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Abstract

Suppose a Poisson process is observed on the unit interval. The scan statistic is defined as the maximum number of events observed as a window of fixed width is moved across the interval, and the distribution under homogeneity has been widely studied. Frequently, we may not wish to specify the window width in advance but to consider scan statistics with varying window widths. We propose a modification of the scan statistic based on a likelihood ratio criterion. This leads to a boundary-crossing problem for a two-dimensional random field, which we approximate using a large-deviation scaling under homogeneity. Similar results are obtained for Poisson processes observed in two dimensions. Numerical computations and simulations are used to illustrate the accuracy of the approximations.

Keywords

BOUNDARY CROSSINGCHANGE POINTSPOISSON PROCESSRANDOM FIELDSPATIAL STATISTICS
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 4 , December 1991 , pp. 751 - 771
Copyright
Copyright © Applied Probability Trust 1991 

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