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On the asymptotic distribution of the discrete scan statistic

Part of: Limit theorems Distribution theory

Published online by Cambridge University Press:  14 July 2016

Michael V. Boutsikas  and
Markos V. Koutras
Michael V. Boutsikas*
Affiliation:
University of Piraeus
Markos V. Koutras*
Affiliation:
University of Piraeus
*
∗Postal address: Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou Street, 185 34 Piraeus, Greece.
∗Postal address: Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou Street, 185 34 Piraeus, Greece.
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Abstract

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The discrete scan statistic in a binary (0-1) sequence of n trials is defined as the maximum number of successes within any k consecutive trials (n and k, n ≥ k, being two positive integers). It has been used in many areas of science (quality control, molecular biology, psychology, etc.) to test the null hypothesis of uniformity against a clustering alternative. In this article we provide a compound Poisson approximation and subsequently use it to establish asymptotic results for the distribution of the discrete scan statistic as n, k → ∞ and the success probability of the trials is kept fixed. An extreme value theorem is also provided for the celebrated Erdős-Rényi statistic.

Keywords

Discrete scan statisticcompound Poisson approximationrandomness testErdős-Rényi statisticKolmogorov distanceextreme value theorem

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic) 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory 60F10: Large deviations
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 1137 - 1154
Copyright
© Applied Probability Trust 2006 

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