IDEAS home Printed from https://ideas.repec.org/a/spr/metrik/v77y2014i7p895-920.html
   My bibliography  Save this article

On sooner and later waiting time distributions associated with simple patterns in a sequence of bivariate trials

Author

Listed:
  • Kiyoshi Inoue
  • Sigeo Aki

Abstract

In this article, we study sooner/later waiting time problems for simple patterns in a sequence of bivariate trials. The double generating functions of the sooner/later waiting times for the simple patterns are expressed in terms of the double generating functions of the numbers of occurrences of the simple patterns. Effective computational tools are developed for the evaluation of the waiting time distributions along with some examples. The results presented here provide perspectives on the waiting time problems arising from bivariate trials and extend a framework for studying the exact distributions of patterns. Finally, some examples are given in order to illustrate how our theoretical results are employed for the investigation of the waiting time problems for simple patterns. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Kiyoshi Inoue & Sigeo Aki, 2014. "On sooner and later waiting time distributions associated with simple patterns in a sequence of bivariate trials," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(7), pages 895-920, October.
    • Handle: RePEc:spr:metrik:v:77:y:2014:i:7:p:895-920
    DOI: 10.1007/s00184-013-0478-0
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00184-013-0478-0
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s00184-013-0478-0?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. Kiyoshi Inoue & Sigeo Aki & Balakrishnan Narayanaswamy, 2013. "Generating Functions of Waiting Times and Numbers of Visits for Random Walks on Graphs," Methodology and Computing in Applied Probability, Springer, vol. 15(2), pages 349-362, June.
    2. S. Robin & J.-J. Daudin, 2001. "Exact Distribution of the Distances between Any Occurrences of a Set of Words," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(4), pages 895-905, December.
    3. Sigeo Aki & Katuomi Hirano, 1999. "Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(1), pages 17-29, March.
    4. Sigeo Aki & Katuomi Hirano, 2013. "Coupon collector’s problems with statistical applications to rankings," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(3), pages 571-587, June.
    5. Kiyoshi Inoue & Sigeo Aki, 2009. "On waiting time distributions associated with compound patterns in a sequence of multi-state trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(2), pages 499-516, June.
    6. Ebneshahrashoob, M. & Sobel, Milton, 1990. "Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas," Statistics & Probability Letters, Elsevier, vol. 9(1), pages 5-11, January.
    7. Sobel, Milton & Frankowski, Krzysztof, 1994. "A duality theorem for solving multiple-player multivariate hypergeometric problems," Statistics & Probability Letters, Elsevier, vol. 20(2), pages 155-162, May.
    8. Marshall, Albert W. & Olkin, Ingram, 1990. "Bivariate distributions generated from Pólya-Eggenberger urn models," Journal of Multivariate Analysis, Elsevier, vol. 35(1), pages 48-65, October.
    9. Demetrios Antzoulakos & Stathis Chadjiconstantinidis, 2001. "Distributions of Numbers of Success Runs of Fixed Length in Markov Dependent Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 599-619, September.
    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. Dheeraj Sharma & Mir Ghulam Talpur, 2016. "An Examination of One Dimension Marginal Distributions: Selling and Non - Selling Activities of a Salesperson," Working Papers id:10970, eSocialSciences.
    2. Sharma, Dheeraj & Talpur, Mir Ghulam Haider, 2016. "An Examination of One Dimension Marginal Distributions: Selling and Non-selling Activities of a Salesperson," IIMA Working Papers WP2016-03-48, Indian Institute of Management Ahmedabad, Research and Publication Department.
    3. Inoue, Kiyoshi & Aki, Sigeo, 2010. "On the conditional and unconditional distributions of the number of success runs on a circle with applications," Statistics & Probability Letters, Elsevier, vol. 80(9-10), pages 874-885, May.
    4. Kiyoshi Inoue & Sigeo Aki, 2018. "Joint distributions of numbers of runs of specified lengths on directed trees," Statistical Papers, Springer, vol. 59(1), pages 249-269, March.
    5. Chang, Yung-Ming, 2005. "Distribution of waiting time until the rth occurrence of a compound pattern," Statistics & Probability Letters, Elsevier, vol. 75(1), pages 29-38, November.
    6. Kim, Bara & Kim, Jeongsim, 2019. "Sooner waiting time problems in a sequence of multi-state trials with random rewards," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 171-179.
    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. 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.
    10. Ana Helena Tavares & Jakob Raymaekers & Peter J. Rousseeuw & Paula Brito & Vera Afreixo, 2020. "Clustering genomic words in human DNA using peaks and trends of distributions," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 14(1), pages 57-76, March.
    11. 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.
    12. D. Antzoulakos & S. Bersimis & M. Koutras, 2003. "On the distribution of the total number of run lengths," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(4), pages 865-884, December.
    13. Kiyoshi Inoue, 2004. "Joint distributions associated with patterns, successes and failures in a sequence of multi-state trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(1), pages 143-168, March.
    14. Sigeo Aki, 2012. "Statistical modeling for discrete patterns in a sequence of exchangeable trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(3), pages 633-655, June.
    15. 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.
    16. Sankaran, P.G. & Midhu, N.N., 2010. "On waiting time distributions for patterns in a sequence of multistate trials," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1798-1805, December.
    17. 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.
    18. 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.
    19. 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.
    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:metrik:v:77:y:2014:i:7:p:895-920. 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.