IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i4p576-d344875.html
   My bibliography  Save this article

One Dimensional Discrete Scan Statistics for Dependent Models and Some Related Problems

Author

Listed:
  • Alexandru Amarioarei

    (Faculty of Mathematics and Computer Science, University of Bucharest, 010014 Bucharest, Romania
    National Institute of Research and Development for Biological Sciences, 060031 Bucharest, Romania)

  • Cristian Preda

    (Laboratoire de Mathématiques Paul Painlevé, University of Lille, 59655 Villeneuve d’Ascq, France
    Biostatistics Department, Delegation for Clinical Research and Innovation, Lille Catholic Hospitals, GHICL, 59462 Lomme, France
    Institute of Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania
    Inria Lille Nord-Europe, MODAL, 59655 Villeneuve d’Ascq, France)

Abstract

The one dimensional discrete scan statistic is considered over sequences of random variables generated by block factor dependence models. Viewed as a maximum of an 1-dependent stationary sequence, the scan statistics distribution is approximated with accuracy and sharp bounds are provided. The longest increasing run statistics is related to the scan statistics and its distribution is studied. The moving average process is a particular case of block factor and the distribution of the associated scan statistics is approximated. Numerical results are presented.

Suggested Citation

  • Alexandru Amarioarei & Cristian Preda, 2020. "One Dimensional Discrete Scan Statistics for Dependent Models and Some Related Problems," Mathematics, MDPI, vol. 8(4), pages 1-11, April.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:4:p:576-:d:344875
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/4/576/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/4/576/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Jie Chen & Joseph Glaz, 2016. "Scan statistics for monitoring data modeled by a negative binomial distribution," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(6), pages 1632-1642, March.
    2. Masayuki Uchida, 1998. "On Generating Functions of Waiting Time Problems for Sequence Patterns of Discrete Random Variables," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(4), pages 655-671, December.
    3. Michael V. Boutsikas & Markos V. Koutras, 2000. "Reliability Approximation for Markov Chain Imbeddable Systems," Methodology and Computing in Applied Probability, Springer, vol. 2(4), pages 393-411, December.
    4. Morteza Ebneshahrashoob & Tangan Gao & Mengnien Wu, 2005. "An Efficient Algorithm for Exact Distribution of Discrete Scan Statistics," Methodology and Computing in Applied Probability, Springer, vol. 7(4), pages 459-471, December.
    5. Shi, Jianxin & Siegmund, David & Yakir, Benny, 2007. "Importance Sampling for Estimating p Values in Linkage Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 929-937, September.
    6. Frolov, Andrei N. & Martikainen, Alexander I., 1999. "On the length of the longest increasing run in d," Statistics & Probability Letters, Elsevier, vol. 41(2), pages 153-161, January.
    7. Haiman, George & Preda, Cristian, 2013. "One dimensional scan statistics generated by some dependent stationary sequences," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1457-1463.
    8. Nathalie Mitton & Katy Paroux & Bruno Sericola & Sébastien Tixeuil, 2010. "Ascending Runs in Dependent Uniformly Distributed Random Variables: Application to Wireless Networks," Methodology and Computing in Applied Probability, Springer, vol. 12(1), pages 51-62, March.
    9. Joseph Glaz & Joseph Naus & Xiao Wang, 2012. "Approximations and Inequalities for Moving Sums," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 597-616, September.
    10. Wang, Xiao & Zhao, Bo & Glaz, Joseph, 2014. "A multiple window scan statistic for time series models," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 196-203.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Qianzhu Wu & Joseph Glaz, 2021. "Scan Statistics for Normal Data with Outliers," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 429-458, March.
    2. Qianzhu Wu & Joseph Glaz, 2019. "Robust Scan Statistics for Detecting a Local Change in Population Mean for Normal Data," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 295-314, March.
    3. Jack Noonan & Anatoly Zhigljavsky, 2021. "Approximations for the Boundary Crossing Probabilities of Moving Sums of Random Variables," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 873-892, September.
    4. Wang, Xiao & Zhao, Bo & Glaz, Joseph, 2014. "A multiple window scan statistic for time series models," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 196-203.
    5. Jie Chen & Thomas Ferguson & Paul Jorgensen, 2020. "Using Scan Statistics for Cluster Detection: Recognizing Real Bandwagons," Methodology and Computing in Applied Probability, Springer, vol. 22(4), pages 1481-1491, December.
    6. Anat Reiner-Benaim, 2016. "Scan Statistic Tail Probability Assessment Based on Process Covariance and Window Size," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 717-745, September.
    7. Haiman, George & Preda, Cristian, 2013. "One dimensional scan statistics generated by some dependent stationary sequences," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1457-1463.
    8. Zhao, Bo & Glaz, Joseph, 2016. "Scan statistics for detecting a local change in variance for normal data with unknown population variance," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 137-145.
    9. Nathalie Mitton & Katy Paroux & Bruno Sericola & Sébastien Tixeuil, 2010. "Ascending Runs in Dependent Uniformly Distributed Random Variables: Application to Wireless Networks," Methodology and Computing in Applied Probability, Springer, vol. 12(1), pages 51-62, March.
    10. Lloyd, Chris J., 2012. "Computing highly accurate or exact P-values using importance sampling," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1784-1794.
    11. Vladimir Pozdnyakov, 2008. "On occurrence of subpattern and method of gambling teams," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(1), pages 193-203, March.
    12. Sabine Mercier & Grégory Nuel, 2022. "Duality Between the Local Score of One Sequence and Constrained Hidden Markov Model," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1411-1438, September.
    13. Vladimir Pozdnyakov & Joseph Glaz & Martin Kulldorff & J. Steele, 2005. "A martingale approach to scan statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(1), pages 21-37, March.
    14. Bo Zhao & Joseph Glaz, 2016. "Scan Statistics for Detecting a Local Change in Variance for Normal Data with Known Variance," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 563-573, June.
    15. Frolov, Andrei & Martikainen, Alexander & Steinebach, Josef, 2000. "Strong laws for the maximal gain over increasing runs," Statistics & Probability Letters, Elsevier, vol. 50(3), pages 305-312, November.
    16. Hisashi Yamamoto & Tomoaki Akiba, 2005. "Evaluating methods for the reliability of a large 2‐dimensional rectangular k‐within‐consecutive‐(r, s)‐out‐of‐(m, n):F system," Naval Research Logistics (NRL), John Wiley & Sons, vol. 52(3), pages 243-252, April.
    17. Frolov, Andrei N., 2003. "On asymptotics of the maximal gain without losses," Statistics & Probability Letters, Elsevier, vol. 63(1), pages 13-23, May.
    18. Dafnis, Spiros D. & Makri, Frosso S. & Philippou, Andreas N., 2019. "The reliability of a generalized consecutive system," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 186-193.
    19. G. Nuel, 2019. "Moments of the Count of a Regular Expression in a Heterogeneous Random Sequence," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 875-887, September.
    20. Morteza Ebneshahrashoob & Tangan Gao & Mengnien Wu, 2005. "An Efficient Algorithm for Exact Distribution of Discrete Scan Statistics," Methodology and Computing in Applied Probability, Springer, vol. 7(4), pages 459-471, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:8:y:2020:i:4:p:576-:d:344875. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.