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Distributions of numbers of runs and scans on directed acyclic graphs with generation

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  • Kiyoshi Inoue
  • Sigeo Aki

Abstract

In this paper, we introduce a class of a directed acyclic graph on the assumption that the collection of random variables indexed by the vertices has a Markov property. We present a flexible approach for the study of the exact distributions of runs and scans on the directed acyclic graph by extending the method of conditional probability generating functions. The results presented here provide a wide framework for developing the exact distribution theory of runs and scans on the graphical models. We also show that our theoretical results can easily be carried out through some computer algebra systems and give some numerical examples in order to demonstrate the feasibility of our theoretical results. As applications, two special reliability systems are considered, which are closely related to our general results. Finally, we address the parameter estimation in the distributions of runs and scans. Copyright Springer-Verlag 2013

Suggested Citation

  • 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.
    • Handle: RePEc:spr:compst:v:28:y:2013:i:3:p:1133-1150
    DOI: 10.1007/s00180-012-0349-6
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Ling, K. D., 1988. "On binomial distributions of order k," Statistics & Probability Letters, Elsevier, vol. 6(4), pages 247-250, March.
    4. Sigeo Aki & Katuomi Hirano, 1996. "Lifetime distribution and estimation problems of consecutive-k-out-of-n:F systems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(1), pages 185-199, March.
    5. 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.
    6. M. Koutras & V. Alexandrou, 1995. "Runs, scans and URN model distributions: A unified Markov chain approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(4), pages 743-766, December.
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    Cited by:

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    2. Dudin, Mikhail & Sekerin, Vladimir & Parfenova, Maria & Babishin, Vladimir & Yurkevich, Evgeny, 2014. "Methodology Making Management Decisions Based on a Modified Ramsey Model," Published Papers dud11, Russian Presidential Academy of National Economy and Public Administration.

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