IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v22y2020i3d10.1007_s11009-019-09743-3.html
   My bibliography  Save this article

On the Distribution of the Number of Success Runs in a Continuous Time Markov Chain

Author

Listed:
  • Boutsikas V. Michael

    (University of Piraeus)

  • Vaggelatou Eutichia

    (National and Kapodistrian University of Athens)

Abstract

We propose a continuous-time adaptation of the well-known concept of success runs by considering a marked point process with two types of marks (success-failure) that appear according to an appropriate continuous-time Markov chain. By constructing a bivariate imbedded process (consisting of a run-counting and a phase process), we offer recursive formulas and generating functions for the distribution of the number of runs and the waiting time until the appearance of the n-th success run. We investigate the three most popular counting schemes: (i) overlapping runs of length k, (ii) non-overlapping runs of length k and (iii) runs of length at least k. We also present examples of applications regarding: the total penalty cost in a maintenance reliability system, the number of risky situations in a non-life insurance portfolio and the number of runs of increasing (or decreasing) asset price movements in high-frequency financial data.

Suggested Citation

  • Boutsikas V. Michael & Vaggelatou Eutichia, 2020. "On the Distribution of the Number of Success Runs in a Continuous Time Markov Chain," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 969-993, September.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:3:d:10.1007_s11009-019-09743-3
    DOI: 10.1007/s11009-019-09743-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-019-09743-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-019-09743-3?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Binswanger, K. & Embrechts, P., 1994. "Longest runs in coin tossing," Insurance: Mathematics and Economics, Elsevier, vol. 15(2-3), pages 139-149, December.
    2. Wu, Tung-Lung & Glaz, Joseph, 2015. "A new adaptive procedure for multiple window scan statistics," Computational Statistics & Data Analysis, Elsevier, vol. 82(C), pages 164-172.
    3. Frosso S. Makri & Zaharias M. Psillakis, 2011. "On Success Runs of Length Exceeded a Threshold," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 269-305, June.
    4. Serkan Eryilmaz, 2018. "Stochastic Ordering Among Success Runs Statistics in a Sequence of Exchangeable Binary Trials," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 563-573, June.
    5. Boutsikas, M. V. & Koutras, M. V., 2002. "Modeling claim exceedances over thresholds," Insurance: Mathematics and Economics, Elsevier, vol. 30(1), pages 67-83, February.
    6. Frosso S. Makri & Zaharias M. Psillakis, 2016. "On runs of ones defined on a q-sequence of binary trials," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(5), pages 579-602, July.
    7. M. Koutras & V. Alexandrou, 1995. "Runs, scans and URN model distributions: A unified Markov chain approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(4), pages 743-766, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. He Yi & Lirong Cui & Narayanaswamy Balakrishnan, 2022. "On the Derivative Counting Processes of First- and Second-order Aggregated Semi-Markov Systems," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1849-1875, September.

    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. Spiros D. Dafnis & Frosso S. Makri, 2023. "Distributions Related to Weak Runs With a Minimum and a Maximum Number of Successes: A Unified Approach," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-24, March.
    2. Spiros D. Dafnis & Frosso S. Makri, 2022. "Weak runs in sequences of binary trials," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(5), pages 573-603, July.
    3. Markos V. Koutras & Demetrios P. Lyberopoulos, 2018. "Asymptotic results for jump probabilities associated to the multiple scan statistic," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(5), pages 951-968, October.
    4. Frosso Makri & Zaharias Psillakis, 2011. "On runs of length exceeding a threshold: normal approximation," Statistical Papers, Springer, vol. 52(3), pages 531-551, August.
    5. Spiros D. Dafnis & Frosso S. Makri & Markos V. Koutras, 2021. "Generalizations of Runs and Patterns Distributions for Sequences of Binary Trials," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 165-185, March.
    6. Eryilmaz, Serkan, 2018. "On success runs in a sequence of dependent trials with a change point," Statistics & Probability Letters, Elsevier, vol. 132(C), pages 91-98.
    7. Sotirios Bersimis & Athanasios Sachlas & Pantelis G. Bagos, 2017. "Discriminating membrane proteins using the joint distribution of length sums of success and failure runs," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 26(2), pages 251-272, June.
    8. Vasileios M. Koutras & Markos V. Koutras & Spiros D. Dafnis, 2022. "A Family of Induced Distributions," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1833-1848, September.
    9. Frosso S. Makri & Zaharias M. Psillakis, 2011. "On Success Runs of Length Exceeded a Threshold," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 269-305, June.
    10. Boutsikas, M. V. & Koutras, M. V., 2002. "Modeling claim exceedances over thresholds," Insurance: Mathematics and Economics, Elsevier, vol. 30(1), pages 67-83, February.
    11. Serkan Eryilmaz, 2005. "On the distribution and expectation of success runs in nonhomogeneous Markov dependent trials," Statistical Papers, Springer, vol. 46(1), pages 117-128, January.
    12. Koutras, Vasileios M. & Koutras, Markos V. & Yalcin, Femin, 2016. "A simple compound scan statistic useful for modeling insurance and risk management problems," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 202-209.
    13. Lou, W. Y. Wendy, 2003. "The exact distribution of the k-tuple statistic for sequence homology," Statistics & Probability Letters, Elsevier, vol. 61(1), pages 51-59, January.
    14. Sigeo Aki & Katuomi Hirano, 2002. "On Waiting Time for Reversed Patterns in Random Sequences," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(4), pages 713-718, December.
    15. Stathis Chadjiconstantinidis & Serkan Eryilmaz, 2023. "Computing waiting time probabilities related to $$ (k_{1},k_{2},\ldots ,k_{l})$$ ( k 1 , k 2 , … , k l ) pattern," Statistical Papers, Springer, vol. 64(5), pages 1373-1390, October.
    16. Shinde, R.L. & Kotwal, K.S., 2006. "On the joint distribution of runs in the sequence of Markov-dependent multi-state trials," Statistics & Probability Letters, Elsevier, vol. 76(10), pages 1065-1074, May.
    17. Kiyoshi Inoue & Sigeo Aki, 2013. "Distributions of numbers of runs and scans on directed acyclic graphs with generation," Computational Statistics, Springer, vol. 28(3), pages 1133-1150, June.
    18. Sigeo Aki, 1999. "Distributions of Runs and Consecutive Systems on Directed Trees," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(1), pages 1-15, March.
    19. Frosso S. Makri & Zaharias M. Psillakis, 2016. "On runs of ones defined on a q-sequence of binary trials," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(5), pages 579-602, July.
    20. Serkan Eryilmaz, 2018. "Stochastic Ordering Among Success Runs Statistics in a Sequence of Exchangeable Binary Trials," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 563-573, June.

    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:spr:metcap:v:22:y:2020:i:3:d:10.1007_s11009-019-09743-3. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.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.