Skip to main content
SpringerLink
Log in

1-Dependent Stationary Sequences and Applications to Scan Statistics

  • Chapter
  • First Online:
Scan Statistics

Part of the book series: Statistics for Industry and Technology ((SIT))

Abstract

A new method of estimating the distribution function of scan statistics was presented and studied by the authors in a series of papers. This method is based on the application of some results concerning the distribution function of the partial maximum sequence generated by a 1-dependent stationary sequence. We present a review of our results and compare the method with other existing methods.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aldous, D. (1989). Probability Approximation via the Poisson Clumping Heuristic, Springer-Verlag, New York.

    Google Scholar 

  2. Alm, S.E. (1997). On the distribution of scan statistics of two-dimensional Poisson processes, Advances Applied Probability, 29, 1–18.

    Article  MATH  MathSciNet  Google Scholar 

  3. Bateman, G.I. (1948). On the power function of the longest run as a test for randomness in a sequence of alternatives, Biometrika, 35, 97–112.

    MATH  MathSciNet  Google Scholar 

  4. Boutsikas, M. and Koutras, M. (2003). Bounds for the distribution of two dimensional binary scan statistics, Probability in the Engineering and Information Sciences, 17, 509–525.

    MATH  MathSciNet  Google Scholar 

  5. Chen, J. and Glaz, J. (1996). Two-dimensional discrete scan statistics, Statistics and Probability letters, 31, 59–68.

    Article  MATH  MathSciNet  Google Scholar 

  6. Fu, J.C. (2001). Distribution of the scan statistic for a sequence of bistate trials, Journal of Applied Probability, 38, 4, 908–916.

    Article  MATH  MathSciNet  Google Scholar 

  7. Fu, J.C., Wang, L. and Lou, W. (2003). On exact and large deviation approximation for the distribution of the longest run in a sequence of two-state Markov dependent trials, Journal of Applied Probability, 40, 2, 346–360.

    Article  MATH  MathSciNet  Google Scholar 

  8. Glaz, J. and Naus, J.I. (1978). Multiple coverage on the line, Annals of Probability 7, 900–906.

    Google Scholar 

  9. Glaz, J., Naus, J. and Wallenstein, S. (2001). Scan Statistics, Springer Series in Statistics, Springer-Verlag, New York.

    MATH  Google Scholar 

  10. Haiman, G. (1999). First passage time for some stationary processes, Stochastic Processes and Their Applications, 80, 231–248.

    Article  MATH  MathSciNet  Google Scholar 

  11. Haiman, G. (2000). Estimating the distribution of scan statistics with high precision, Extremes, 3:4, 349–361.

    Google Scholar 

  12. Haiman, G. (2007). Estimating the distribution of one-dimensional discrete scan statistics viewed as extremes of 1-dependent stationary sequences, Journal of Statistical Planning and Inference, 137:3, 821–828.

    Google Scholar 

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

    Google Scholar 

  14. Haiman, G. and Preda, C. (2002). A new method for estimating the distribution of scan statistics for a two-dimensional Poisson process, Methodology and Computing in Applied Probability, 4, 393–407.

    Article  MATH  MathSciNet  Google Scholar 

  15. Haiman, G. and Preda, C. (2006). Estimation for the distribution of two-dimensional discrete scan statistics, Methodology and Computing in Applied Probability, 8, 373–382.

    Article  MATH  MathSciNet  Google Scholar 

  16. Huntington, R.J. (1974). Distributions and expectations for clusters in continuous and discrete cases, with applications, Ph.D. Thesis, Rutgers University.

    Google Scholar 

  17. Huntington, R.J. (1978). Distribution of the minimum number of points in a scanning interval on the line. Stochastic Processes and Their Applications, 7, 73–78.

    Article  MATH  MathSciNet  Google Scholar 

  18. Huntington, R.J. and Naus, J.I. (1975). A simpler expression for kth nearest-neighbor coincidence probabilities, Annals of Probability, 3, 894–896.

    Article  MATH  MathSciNet  Google Scholar 

  19. Janson, S. (1984). Bounds on the distribution of extremal values of a scanning process, Stochastic Processes and Their Applications, 18, 313–328.

    Article  MATH  MathSciNet  Google Scholar 

  20. Karwe, V.V. (1993). The distribution of the supremum of integer moving average processes with applications to the maximum net charge in DNA sequences, Ph.D. Thesis, Rutgers University.

    Google Scholar 

  21. Lou, W. (1996). On runs and longest run tests: a method of finite Markov chain imbedding, J. Amer. Statist. Assoc. 91, 1595–1601.

    Article  MATH  MathSciNet  Google Scholar 

  22. Naus, J.I. (1965). A power comparison of two tests of non-random clustering, Technometrics, 8, 493–517.

    Article  MathSciNet  Google Scholar 

  23. Naus, J.I. (1982). Approximations for distributions of scan statistics, Journal of the American Statistical Association, 77, 177–183.

    Article  MATH  MathSciNet  Google Scholar 

  24. Neff, N. (1978) Piecewise polynomials for the probability of clustering on the unit interval, Unpublished PhD. dissertation, Rutgers University.

    Google Scholar 

  25. Neff N.D. and Naus, J.I. (1980). The distribution of the size of the maximum cluster points on a line, In IMS Series Selected Tables in Mathematical Statistics, Vol. IV, American Mathematical Society, Providence, RI.

    Google Scholar 

  26. Saperstein, B. (1976). The analysis of attribute moving averages: MILSTD-105D reduced inspection plan, Sixth Conference Stochastic Processes and Applications, Tel Aviv.

    Google Scholar 

  27. Vaggelatou, E. (2003). On the length of the longest run in a multi-state Markov chain, Statistics & Probability Letters, 62:3, 211–221.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Lille 1 Lille, UFR de Mathématiques, France

    George Haiman

  2. Université de Lille 2, Lille, Faculté de Médecine, 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. Dept. Statistics, University of Connecticut, Glenbrook Rd. 215, Storrs, 06269-3120, U.S.A.

    Joseph Glaz

  2. Dept. Statistics, University of Connecticut, Glenbrook Road 215, Storrs, 06269-4120, U.S.A.

    Vladimir Pozdnyakov

  3. Center for Biomathematical Sciences, Mount Sinai School of Medicine, Gustave L Levy Place 1, New York, 10029, U.S.A.

    Sylvan Wallenstein

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Birkhäuser Boston, a part of Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Haiman, G., Preda, C. (2009). 1-Dependent Stationary Sequences and Applications to Scan Statistics. In: Glaz, J., Pozdnyakov, V., Wallenstein, S. (eds) Scan Statistics. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4749-0_8

Download citation

Publish with us

Policies and ethics

Search

Navigation