IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v24y2022i4d10.1007_s11009-022-09948-z.html
   My bibliography  Save this article

Distributions of $$({k}_{1},{k}_{2},\dots ,{k}_{m})$$ ( k 1 , k 2 , ⋯ , k m ) -runs with Multi-state Trials

Author

Listed:
  • Xian Zhao

    (Beijing Institute of Technology)

  • Yanbo Song

    (Beijing Institute of Technology)

  • Xiaoyue Wang

    (Beijing Technology and Business University)

  • Zhiyue Lv

    (AVIC China Aero-Polytechnology Establishment)

Abstract

In this paper, six new $$({k}_{1},{k}_{2},\dots ,{k}_{m})$$ ( k 1 , k 2 , ⋯ , k m ) -runs with multi-state trials are proposed creatively, which can satisfy the practical needs in many fields. The exact distributions of proposed runs are obtained by applying finite Markov chain imbedding approach. This paper not only studies the case of independent identical distribution (i.i.d.) multi-state trials, but also independent non-identical distribution (non-i.i.d.) multi-state trials. Numerical examples have served the purpose to illustrate the effectiveness of the proposed approach. This study is of reference value and application significance for similar runs.

Suggested Citation

  • Xian Zhao & Yanbo Song & Xiaoyue Wang & Zhiyue Lv, 2022. "Distributions of $$({k}_{1},{k}_{2},\dots ,{k}_{m})$$ ( k 1 , k 2 , ⋯ , k m ) -runs with Multi-state Trials," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2689-2702, December.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:4:d:10.1007_s11009-022-09948-z
    DOI: 10.1007/s11009-022-09948-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-022-09948-z
    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/s11009-022-09948-z?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. A. N. Kumar & N. S. Upadhye, 2019. "Generalizations of distributions related to ( $$k_1,k_2$$ k 1 , k 2 )-runs," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(2), pages 249-268, March.
    2. Markos V. Koutras & Serkan Eryilmaz, 2017. "Compound Geometric Distribution of Order k," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 377-393, June.
    3. Philippou, Andreas N. & Georghiou, Costas & Philippou, George N., 1983. "A generalized geometric distribution and some of its properties," Statistics & Probability Letters, Elsevier, vol. 1(4), pages 171-175, June.
    4. K. Kotwal & R. Shinde, 2006. "Joint distributions of runs in a sequence of higher-order two-state Markov trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(3), pages 537-554, September.
    5. A. N. Kumar & N. S. Upadhye, 2017. "On perturbations of Stein operator," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(18), pages 9284-9302, September.
    6. EryIlmaz, Serkan, 2008. "Distribution of runs in a sequence of exchangeable multi-state trials," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1505-1513, September.
    7. Narayanaswamy Balakrishnan & Alexei Stepanov, 2013. "Runs Based on Records: Their Distributional Properties and an Application to Testing for Dispersive Ordering," Methodology and Computing in Applied Probability, Springer, vol. 15(3), pages 583-594, September.
    8. Qing Han & Sigeo Aki, 1999. "Joint Distributions of Runs in a Sequence of Multi-State Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(3), pages 419-447, September.
    9. 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.
    10. Wang, Xiaoyue & Zhao, Xian & Wang, Siqi & Sun, Leping, 2020. "Reliability and maintenance for performance-balanced systems operating in a shock environment," Reliability Engineering and System Safety, Elsevier, vol. 195(C).
    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. Kirtee Kamalja, 2014. "On the joint distribution of success runs of several lengths in the sequence of MBT and its applications," Statistical Papers, Springer, vol. 55(4), pages 1179-1206, November.
    2. Kumar, Amit N. & Kumar, Poleen, 2024. "A negative binomial approximation to the distribution of the sum of maxima of indicator random variables," Statistics & Probability Letters, Elsevier, vol. 208(C).
    3. 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.
    4. Spiros Dafnis & Andreas Philippou & Demetrios Antzoulakos, 2012. "Distributions of patterns of two successes separated by a string of k-2 failures," Statistical Papers, Springer, vol. 53(2), pages 323-344, May.
    5. Qing Han & Sigeo Aki, 2000. "Waiting Time Problems in a Two-State Markov Chain," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(4), pages 778-789, December.
    6. Upadhye, N.S. & Kumar, A.N., 2018. "Pseudo-binomial approximation to (k1,k2)-runs," Statistics & Probability Letters, Elsevier, vol. 141(C), pages 19-30.
    7. Kiyoshi Inoue, 2021. "On Homogeneous Multivariate Distributions in Random Occupancy Models and Their Applications," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 1129-1153, September.
    8. 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.
    9. Zhao, Xian & He, Zongda & Wu, Yaguang & Qiu, Qingan, 2022. "Joint optimization of condition-based performance control and maintenance policies for mission-critical systems," Reliability Engineering and System Safety, Elsevier, vol. 226(C).
    10. 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.
    11. Shmerling, Efraim, 2013. "A range reduction method for generating discrete random variables," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1094-1099.
    12. Dui, Hongyan & Chen, Shuanshuan & Wang, Jia, 2021. "Failure-oriented maintenance analysis of nodes and edges in network systems," Reliability Engineering and System Safety, Elsevier, vol. 215(C).
    13. Sigeo Aki & Katuomi Hirano, 1994. "Distributions of numbers of failures and successes until the first consecutivek successes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(1), pages 193-202, March.
    14. Pinciroli, Luca & Baraldi, Piero & Zio, Enrico, 2023. "Maintenance optimization in industry 4.0," Reliability Engineering and System Safety, Elsevier, vol. 234(C).
    15. A. N. Kumar & N. S. Upadhye, 2019. "Generalizations of distributions related to ( $$k_1,k_2$$ k 1 , k 2 )-runs," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(2), pages 249-268, March.
    16. Zhao, Xian & Han, He & Jiao, Chunhui & Qiu, Qingan, 2024. "Reliability modeling of k-out-of-n: F balanced systems with common bus performance sharing," Reliability Engineering and System Safety, Elsevier, vol. 248(C).
    17. Zhao, Xian & Qi, Xin & Wang, Xiaoyue, 2023. "Reliability assessment for coherent systems operating under a generalized mixed shock model with multiple change points of the environment," Reliability Engineering and System Safety, Elsevier, vol. 239(C).
    18. Drekic, Steve & Spivey, Michael Z., 2021. "On the number of trials needed to obtain k consecutive successes," Statistics & Probability Letters, Elsevier, vol. 176(C).
    19. Kumar C. Satheesh & Nair B. Unnikrishnan, 2013. "Order k Version of the Alternative Hyper-Poisson Distribution," Stochastics and Quality Control, De Gruyter, vol. 28(2), pages 97-104, December.
    20. Muselli, Marco, 1996. "Simple expressions for success run distributions in bernoulli trials," Statistics & Probability Letters, Elsevier, vol. 31(2), pages 121-128, 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:spr:metcap:v:24:y:2022:i:4:d:10.1007_s11009-022-09948-z. 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.