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Distribution of the scan statistic for a sequence of bistate trials

Part of: Markov processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

James C. Fu
James C. Fu*
Affiliation:
University of Manitoba
*
∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2. Email address: fu@ccu.umanitoba.ca
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Abstract

Let be the scan statistic of window size r for a sequence of n bistate trials . The scan statistic S n (r) has been successfully used in various fields of applied probability and statistics, and its distribution has been studied extensively in the literature. Currently, all existing formulae for the distribution of S n (r) are rather complex, and they can only be numerically implemented when is a sequence of Bernoulli trials, the window size r is less than 20 and the length of the sequence n is not too large. Hence, these formulae have been limiting the practical applications of the scan statistic. In this article, we derive a simple and effective formula for the distribution of S n (r) via the finite Markov chain embedding technique to overcome some of the limitations of the existing complex formulae. This new formula can be applied when is either a sequence of Bernoulli trials or a sequence of Markov dependent bistate trials. Selected numerical examples are given to illustrate our results.

Keywords

Compound patternwaiting time distributionfinite Markov chain embeddingtransition probability matrix

MSC classification

Primary: 60E05: Distributions
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 908 - 916
Copyright
Copyright © by the Applied Probability Trust 2001 

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Footnotes

This work was supported in part by Grant A-9216 of the Natural Science and Engineering Research Council of Canada.

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