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Discrete Scan Statistics Generated by Exchangeable Binary Trials

Part of: Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Serkan Eryilmaz
Serkan Eryilmaz*
Affiliation:
Izmir University of Economics
*
∗Current address: Department of Industrial Engineering, Atilim University, 06836 Incek, Ankara, Turkey. Email address: seryilmaz@atilim.edu.tr
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Abstract

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Let {Xi}i=1n be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable Sn,m to be the maximum number of 1s within any m consecutive trials in {Xi}i=1n. The random variable Sn,m is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {Xi}i=1n consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of Sn,m for 2m ≥ n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.

Keywords

Bernoulli sequenceexchangeable binary trialreliabilityscan statistics

MSC classification

Secondary: 60E05: Distributions 62E15: Exact distribution theory
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1084 - 1092
Copyright
Copyright © Applied Probability Trust 2010 

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