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Demimartingale Approaches for Scan Statistics

  • Markos V. KoutrasEmail author
  • Demetrios P. Lyberopoulos
Living reference work entry

Abstract

Scan statistics are defined as random variables enumerating the moving windows in a sequence of binary outcome trials which contain a prescribed number of successes. The main objective of this contribution is to serve as a self-contained source of some recent results concerning both the simple and the multiple scan statistic. These results are innovative in the sense that they seem to be the first ones on scan statistics that were derived by means of demimartingale techniques. The demimartingale approach motivated also some classification questions for stochastic processes associated with scan statistics. These types of questions and some past results on scan statistics that can be regarded as relevant to the demimartingale approach are also discussed here. In order to illustrate how our results can be implemented in practice, our presentation is enriched with several numerical exhibitions.

Keywords

Scan statistic Multiple scan statistic Demimartingale Demisubmartingale N-demimartingale N-demisupermartingale Bound Asymptotic results Maximal inequalities Binary trial 

Notes

Acknowledgements

D.P.L. would like to dedicate this work in memory of his father Panagiotis (Takis). His support during this research endeavor of D.P.L. is one of the many moving memories which the co-author will always recall, full of love and gratitude!

This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) – Research Funding Program: Aristeia II - Investing in knowledge society through the European Social Fund.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Markos V. Koutras
    • 1
    Email author
  • Demetrios P. Lyberopoulos
    • 1
  1. 1.Department of Statistics and Insurance ScienceUniversity of PiraeusPiraeusGreece

Section editors and affiliations

  • Joseph Glaz
    • 1
  • Markos V. Koutras
    • 2
  1. 1.Department of StatisticsUniversity of ConnecticutStorrsUSA
  2. 2.Dept. of Statistics and Insurance Science, University of PiraeusPiraeusGreece

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