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Waiting Time for an Almost Perfect Run and Applications in Statistical Process Control

Author

Listed:
  • Sotiris Bersimis

    (University of Pireaus)

  • Markos V. Koutras

    (University of Pireaus)

  • George K. Papadopoulos

    (Agricultural University of Athens)

Abstract

A natural and intuitively appealing generalization of the runs principle arises if instead of looking at fixed-length strings with all their positions occupied by successes, we allow the appearance of a small number of failures. Therefore, the focus is on clusters of consecutive trials which contain large proportion of successes. Such a formation is traditionally called “scan” or alternatively, due to the high concentration of successes within it, almost perfect (success) run. In the present paper, we study in detail the waiting time distribution for random variables related to the first occurrence of an almost perfect run in a sequence of Bernoulli trials. Using an appropriate Markov chain embedding approach we present an efficient recursive scheme that permits the construction of the associated transition probability matrix in an algorithmically efficient way. It is worth mentioning that, the suggested methodology, is applicable not only in the case of almost perfect runs, but can tackle the general discrete scan case as well. Two interesting applications in statistical process control are also discussed.

Suggested Citation

  • Sotiris Bersimis & Markos V. Koutras & George K. Papadopoulos, 2014. "Waiting Time for an Almost Perfect Run and Applications in Statistical Process Control," Methodology and Computing in Applied Probability, Springer, vol. 16(1), pages 207-222, March.
    • Handle: RePEc:spr:metcap:v:16:y:2014:i:1:d:10.1007_s11009-012-9307-6
    DOI: 10.1007/s11009-012-9307-6
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    References listed on IDEAS

    as
    1. Markos V. Koutras & Sotirios Bersimis & Demetrios L. Antzoulakos, 2006. "Improving the Performance of the Chi-square Control Chart via Runs Rules," Methodology and Computing in Applied Probability, Springer, vol. 8(3), pages 409-426, September.
    2. Chang, C. J. & Fann, C. S. J. & Chou, W. C. & Lian, I. B., 2003. "On the tail probability of the longest well-matching run," Statistics & Probability Letters, Elsevier, vol. 63(3), pages 267-274, July.
    3. Athanasios C. Rakitzis & Demetrios L. Antzoulakos, 2011. "Chi-square Control Charts with Runs Rules," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 657-669, December.
    4. Han, Qing & Hirano, Katuomi, 2003. "Waiting time problem for an almost perfect match," Statistics & Probability Letters, Elsevier, vol. 65(1), pages 39-49, October.
    5. M. V. Koutras & S. Bersimis & P. E. Maravelakis, 2007. "Statistical Process Control using Shewhart Control Charts with Supplementary Runs Rules," Methodology and Computing in Applied Probability, Springer, vol. 9(2), pages 207-224, June.
    6. Bersimis, Sotiris & Psarakis, Stelios & Panaretos, John, 2006. "Multivariate Statistical Process Control Charts: An Overview," MPRA Paper 6399, University Library of Munich, Germany.
    7. Galit Shmueli, 2002. "System-Wide Probabilities for Systems with Runs and Scans Rules," Methodology and Computing in Applied Probability, Springer, vol. 4(4), pages 409-419, December.
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    Cited by:

    1. 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.
    2. Hsing-Ming Chang & James C. Fu, 2022. "On Distribution and Average Run Length of a Two-Stage Control Process," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2723-2742, December.

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