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On the length and the position of the minimum sequence containing all runs of ones in a Markovian binary sequence

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  • Arapis, Anastasios N.
  • Makri, Frosso S.
  • Psillakis, Zaharias M.

Abstract

Let a sequence of binary (zero–one) trials which forms a nonhomogeneous/homogeneous Markov chain of first order with given initial probability distribution and one-step transition probability matrix. The trials are assumed to be ordered on a line. The statistics denoting the length and the position (starting, ending) of the shortest segment of the sequence containing all runs of ones, are considered. The study concerns with conditional probability distributions of the statistics given that there are at least two runs of ones in the sequence. Two application case studies related to learning experiments and DNA sequences are discussed. Illustrative numerics are presented.

Suggested Citation

  • Arapis, Anastasios N. & Makri, Frosso S. & Psillakis, Zaharias M., 2016. "On the length and the position of the minimum sequence containing all runs of ones in a Markovian binary sequence," Statistics & Probability Letters, Elsevier, vol. 116(C), pages 45-54.
    • Handle: RePEc:eee:stapro:v:116:y:2016:i:c:p:45-54
    DOI: 10.1016/j.spl.2016.03.011
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    References listed on IDEAS

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    1. Frosso Makri & Zaharias Psillakis, 2013. "Exact distributions of constrained (k, ℓ) strings of failures between subsequent successes," Statistical Papers, Springer, vol. 54(3), pages 783-806, August.
    2. Frosso Makri & Zaharias Psillakis, 2011. "On runs of length exceeding a threshold: normal approximation," Statistical Papers, Springer, vol. 52(3), pages 531-551, August.
    3. Koutras, M. V. & Alexandrou, V. A., 1997. "Non-parametric randomness tests based on success runs of fixed length," Statistics & Probability Letters, Elsevier, vol. 32(4), pages 393-404, April.
    4. Eryilmaz, Serkan, 2015. "Discrete time shock models involving runs," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 93-100.
    5. 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.
    6. Adrian Raftery & Simon Tavaré, 1994. "Estimation and Modelling Repeated Patterns in High Order Markov Chains with the Mixture Transition Distribution Model," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 43(1), pages 179-199, March.
    7. EryIlmaz, Serkan, 2011. "Joint distribution of run statistics in partially exchangeable processes," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 163-168, January.
    8. Makri, Frosso S. & Psillakis, Zaharias M. & Arapis, Anastasios N., 2015. "Length of the minimum sequence containing repeats of success runs," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 28-37.
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    Cited by:

    1. Ioannis S. Triantafyllou, 2020. "m -Consecutive- k -out-of- n : F Structures with a Single Change Point," Mathematics, MDPI, vol. 8(12), pages 1-14, December.
    2. Anastasios N. Arapis & Frosso S. Makri & Zaharias M. Psillakis, 2017. "Joint distribution of k-tuple statistics in zero-one sequences of Markov-dependent trials," Journal of Statistical Distributions and Applications, Springer, vol. 4(1), pages 1-13, December.
    3. 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.

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