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Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k

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Abstract

A new distribution called a generalized binomial distribution of order k is defined and some properties are investigated. A class of enumeration schemes for success-runs of a specified length including non-overlapping and overlapping enumeration schemes is rigorously studied. For each nonnegative integer μ less than the specified length of the runs, an enumeration scheme called μ-overlapping way of counting is defined. Let k and ℓ be positive integers satisfying ℓ < k. Based on independent Bernoulli trials, it is shown that the number of (ℓ− 1)-overlapping occurrences of success-run of length k until the n-th overlapping occurrence of success-run of length ℓ follows the generalized binomial distribution of order (k−ℓ). In particular, the number of non-overlapping occurrences of success-run of length k until the n-th success follows the generalized binomial distribution of order (k− 1). The distribution remains unchanged essentially even if the underlying sequence is changed from the sequence of independent Bernoulli trials to a dependent sequence such as higher order Markov dependent trials. A practical example of the generalized binomial distribution of order k is also given.

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References

  • Aki, S. (1985). Discrete distributions of order k on a binary sequence, Ann. Inst. Statist. Math., 37, 205–224.

    Google Scholar 

  • Aki, S. and Hirano, K. (1994). Distributions of numbers of failures and successes until the first consecutive k successes, Ann. Inst. Statist. Math., 46, 193–202.

    Google Scholar 

  • Aki, S. and Hirano, K. (1995). Joint distributions of numbers of success-runs and failures until the first consecutive k successes, Ann. Inst. Statist. Math., 47, 225–235.

    Google Scholar 

  • Aki, S., Balakrishnan, N. and Mohanty, S. G. (1996). Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials, Ann. Inst. Statist. Math., 48, 773–787.

    Google Scholar 

  • Balakrishnan, N., Mohanty, S. G. and Aki, S. (1997). Start-up demonstration tests under Markov dependence model with corrective actions, Ann. Inst. Statist. Math., 49, 155–169.

    Google Scholar 

  • Balasubramanian, K., Viveros, R. and Balakrishnan, N. (1995). Some discrete distributions related to extended Pascal triangles, Fibonacci Quart., 33, 415–425.

    Google Scholar 

  • Chao, M. T., Fu, J. C. and Koutras, M. V. (1995). Survey of reliability studies of consecutive-k-out-of-n:F & related systems, IEEE Transactions on Reliability, 40, 120–127.

    Google Scholar 

  • Ebneshalnashoob, M. and Sobel, M. (1990). Sooner and later problems for Bernoulli trials: frequency and run quotas, Statist. Probab. Lett., 9, 5–11.

    Google Scholar 

  • Hahn, G. J. and Gage, J. B. (1983). Evaluation of a start-up demonstration test, Journal of Quality Technology, 15, 103–106.

    Google Scholar 

  • Hirano, K. (1986). Some properties of the distributions of order k, Fibonacci Numbers and Their Applications (eds. A. N. Philippou, G. E. Bergum and A. F. Horadam), 43–53, Reidel, Dordrecht.

    Google Scholar 

  • Hirano, K. (1994). Consecutive-k-out-of-n:F Systems, Proc. Inst. Statist. Math., 42, 45–61 (in Japanese).

    Google Scholar 

  • Hirano, K., Aki, S. and Uchida, M. (1997). Distributions of numbers of success-runs until the first consecutive k successes in higher order Markov dependent trials, Advances in Combinatorial Methods and Applications to Probability and Statistics (ed. by N. Balakrishnan), 401–410, Birkhäuser, Boston.

    Google Scholar 

  • Johnson, N. L., Kotz, S. and Kemp, A. W. (1992). Univariate Discrete Distributions, Wiley, New York.

    Google Scholar 

  • Ling, K. D. (1988). On binomial distributions of order k, Statist. Probab. Lett., 6, 247–250.

    Google Scholar 

  • Philippou, A. N. and Makri, F. S. (1986). Successes, runs and longest runs, Statist. Probab. Lett., 4, 101–105.

    Google Scholar 

  • Philippou, A. N., Georghiou, C. and Philippou, G. N. (1983). A generalized geometric distribution and some of its properties, Statist. Probab. Lett., 1, 171–175.

    Google Scholar 

  • Viveros, R. and Balakrishnan, N. (1993). Statistical inference from start-up demonstration test data, Journal of Quality Technology, 25, 119–130.

    Google Scholar 

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Authors and Affiliations

  1. Department of Informatics and Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, 560-8531, Japan

    Sigeo Aki

  2. The Institute of Statistical Mathematics, 4-6-7 Minani-Azabu, Minato-ku, Tokyo, 106-8569, Japan

    Katuomi Hirano

Authors
  1. Sigeo Aki
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  2. Katuomi Hirano
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Aki, S., Hirano, K. Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k . Annals of the Institute of Statistical Mathematics 52, 767–777 (2000). https://doi.org/10.1023/A:1017585512412

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  • DOI: https://doi.org/10.1023/A:1017585512412

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