Skip to main content
SpringerLink
Log in

Scan Statistics Viewed as Maximum of 1-Dependent Random Variables

  • Living reference work entry
  • First Online:
Handbook of Scan Statistics

  • 207 Accesses

Abstract

A method of approximating the distribution function of the partial maximum sequence generated by a 1-dependent stationary sequence can be applied to estimate the distribution function of one or multidimensional scan statistics. The method, which provides error bounds for the approximations, was investigated and evaluated in several papers.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  • Aldous D (1989) Probability approximation via the Poisson clumping heuristic. Springer, New York

    Book  Google Scholar 

  • Alm SE (1997) On the distribution of scan statistics of two-dimensional Poisson processes. Adv Appl Probab 29:1–18

    Article  MathSciNet  Google Scholar 

  • Amarioarei A (2012) Approximation for the distribution of extremes of one dependent stationary sequences of random variables. arXiv:1211.5456 [math.PR]

    Google Scholar 

  • Amarioarei A (2014) Approximations for multidimensional discrete scan statistics. Phd Thesis, no. 41498, Université de Lille 1, France. https://alexamarioarei.github.io/Research/docs/Thesis.pdf

  • Amarioarei A, Preda C (2014) Approximation for two-dimensional discrete scan statistic in some block-factor type dependent models. J Stat Plann Inference 151–152:107–120

    Article  MathSciNet  Google Scholar 

  • Amarioarei A, Preda C (2015) Approximation for the distribution of three-dimensional discrete scan statistic. Methodol Comput Appl Probab 17(3):565–578

    Article  MathSciNet  Google Scholar 

  • Boutsikas M, Koutras M (2003) Bounds for the distribution of two dimensional binary scan statistics. Probab Eng Inf Sci 17:509–525

    Article  MathSciNet  Google Scholar 

  • Fu JC (2001) Distribution of the scan statistic for a sequence of bistate trials. J Appl Probab 38: 59–68

    Article  MathSciNet  Google Scholar 

  • Glaz J, Naus J (1991) Tight bounds and approximations for scan statistics probabilities for discrete data. Ann Appl Probab 1:306–318

    Article  MathSciNet  Google Scholar 

  • Glaz J, Naus J, Wallenstein S (2001) Scan statistics. Springer series in statistics, Springer, New York

    Book  Google Scholar 

  • Glaz J, Pozdnyakov V, Wallenstein S (2009) Scan statistics methods and applications. Statistics for industry and technology. Birkhäuser, Boston

    MATH  Google Scholar 

  • Glaz J, Naus J, Wang X (2012) Approximations and inequalities for moving sums. Methodol Comput Appl Probab 14(3):597–613

    Article  MathSciNet  Google Scholar 

  • Haiman G (1999) First passage time for some stationary processes. Stoch Process Appl 80:231–248

    Article  MathSciNet  Google Scholar 

  • Haiman G (2000) Estimating the distribution of scan statistics with high precision. Extremes 3:349–361

    Article  MathSciNet  Google Scholar 

  • Haiman G (2007) Estimation the distribution of one-dimensional discrete scan statistics viewed as extremes of 1-dependent stationary sequences. J Stat Plann Inference 137:821–828

    Article  MathSciNet  Google Scholar 

  • Haiman G, Preda C (2002) A new method for estimating the distribution of scan statistics for a two-dimesional Poisson process. Methodol Comput Appl Probab 4:393–407

    Article  MathSciNet  Google Scholar 

  • Haiman G, Preda C (2006) Estimation for the distribution of two dimensional discrete scan statistics. Methodol Comput Appl Probab 8:373–382

    Article  MathSciNet  Google Scholar 

  • Haiman G, Preda C (2013) One dimensional scan statistics generated by some dependent stationary sequences. Stat Probab Lett 83:1457–1463

    Article  MathSciNet  Google Scholar 

  • Haiman G, Mayeur N, Nevzorov V, Puri ML (1998) Records and 2-block records of 1-dependent stationary sequences under local dependence. Annales de l’Institut Henri Poincaré (B) Probabilité et Statistiques 34(4):481–503

    Article  MathSciNet  Google Scholar 

  • Huntington RJ, Naus JI (1975) A simpler expression for kth nearest-neighbor coincidence probabilities. Ann Probab 3:894–896

    Article  Google Scholar 

  • Karwe VV (1993) The distribution of the supremum of integer moving average processes with applications to maximum net charge in DNA sequences. PhD Thesis, Rutgers University

    Google Scholar 

  • Naiman DQ, Priebe CE (2001) Computing scan statistic p values using importance sampling, with applications to genetics and medical image analysis. J Comput Graph Stat 10:296–328

    Article  MathSciNet  Google Scholar 

  • Naus JI (1965) A power comparison of two tests of non-random clustering. Technometrics 8: 493–517

    MathSciNet  MATH  Google Scholar 

  • Naus JI (1982) Approximations for distributions of scan statistics. J Am Stat Assoc 77:177–183

    Article  MathSciNet  Google Scholar 

  • Neff ND (1978) Piecevise polynomials for the probability of clustering on the unit interval. Unpublished PhD. dissertation, Rutgers University

    Google Scholar 

  • Neff ND, Naus JI (1980) The distribution of the size of the maximum cluster of points on a line. Volume VI in IMS series of selected tables in mathematical statistics. American Mathematical Society, Providence

    Google Scholar 

  • Saperstein B (1976) The analysis of attribute moving averages: MIL-STD-105D reduced inspection plan. In: Sixth conference stochastic processes and applications, Tel Aviv

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Montrouge, France

    George Haiman

  2. Laboratoire Paul Painlevé, Université de Lille, Lille, France

    Cristian Preda

Authors
  1. George Haiman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cristian Preda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George Haiman .

Editor information

Editors and Affiliations

  1. University of Connecticut Dept. Statistics, Storrs, CT, USA

    Joseph Glaz

  2. University of Pireaus Dept. Statistics & Insurance Sciences, Piraeus, Greece

    Markos V. Koutras

Section Editor information

  1. Department of Statistics, University of Connecticut, Storrs, CT, USA

    Joseph Glaz

Rights and permissions

Reprints and permissions

Copyright information

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

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Haiman, G., Preda, C. (2019). Scan Statistics Viewed as Maximum of 1-Dependent Random Variables. In: Glaz, J., Koutras, M. (eds) Handbook of Scan Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8414-1_9-1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4614-8414-1_9-1

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-8414-1

  • Online ISBN: 978-1-4614-8414-1

  • eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation