Skip to main content
SpringerLink
Log in

Joint distributions of numbers of runs of specified lengths on directed trees

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

In this paper, we consider exact joint distributions of numbers of success runs of specified lengths on a Markov directed tree under \(\ell \)-overlapping enumeration schemes. Methods for deriving the probability generating functions of the joint distribution are presented. We extend the exact distribution theory of runs based on sequences to runs based on directed trees. Some numerical results for the run statistics are given in order to illustrate the computational aspects and the feasibility of our theoretical results. The reliability systems and lifetime distributions are considered. Our general results are applied to reliability systems and lifetime distributions. Finally, we discuss parametric estimation problems based on the lifetimes of the systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Aki S (1999) Distributions of runs and consecutive systems on directed trees. Ann Inst Stat Math 51:1–15

    Article  MathSciNet  MATH  Google Scholar 

  • Aki S (2001) Exact reliability and lifetime of consecutive systems. In: Balakrishnan N, Rao CR (eds) Handbook of statistics, vol 20. Elsevier, Columbus, pp 281–300

    Google Scholar 

  • Aki S, Hirano K (1988) Some characteristics of the binomial distribution of order \(k\) and related distributions. In: Matusita K (ed) Proceedings of the Second Pacific Area Statistical Conference on Statistical Theory and Data Analysis II. Amsterdam, pp 211–222

  • Aki S, Hirano K (2000) Numbers of success-runs of specified length until certain stopping time rules and generalized binomial distributions of order \(k\). Ann Inst Stat Math 52:767–777

    Article  MathSciNet  MATH  Google Scholar 

  • Aki S, Inoue K (2012) Exact distributions of the number of pattern occrrences in undirected graphical models. J Jpn Stat Soc 42:59–79

    Article  MATH  Google Scholar 

  • Antzoulakos DL, Chadjiconstantinidis S (2001) Distributions of numbers of success runs of fixed length in Markov dependent trials. Ann Inst Stat Math 53:599–619

    Article  MathSciNet  MATH  Google Scholar 

  • Balakrishnan N, Koutras MV (2002) Runs and scans with applications. Wiley, New York

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Chang GJ, Cui L, Hwang FK (2000) Reliabilities of consecutive-\(k\) systems. Kluwer Academic Publishers, Dordrecht

    MATH  Google Scholar 

  • Chao MT, Fu JC, Koutras MV (1995) Survey of reliability studies of consecutive-\(k\)-out-of-\(n\): F & related systems. IEEE Trans Reliab 40:120–127

    Article  Google Scholar 

  • Cui L, Kuo W, Li J, Xie M (2006) On the dual reliability systems of \((n, f, k)\) and \(\langle n, f, k \rangle \). Stat Probab Lett 76:1081–1088

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Eryilmaz S (2008) Run statistics defined on the multicolor urn model. J Appl Proabab 45(4):1007–1023

    Article  MathSciNet  MATH  Google Scholar 

  • Eryilmaz S (2011) Joint distribution of run statistics in partially exchangeable processes. Stat Probab Lett 81:163–168

    Article  MathSciNet  MATH  Google Scholar 

  • Feller W (1968) An introduction to probability theory and its applications, vol 1, 3rd edn. Wiley, New York

    MATH  Google Scholar 

  • Fu JC (1996) Distribution theory of runs and patterns associated with a sequence of multi-state trials. Stat Sin 6:957–974

    MathSciNet  MATH  Google Scholar 

  • Fu JC, Koutras MV (1994) Distribution theory of runs: a Markov chain approach. J Am Stat Assoc 89:1050–1058

    Article  MathSciNet  MATH  Google Scholar 

  • Fu JC, Lou WYW (2003) Distribution theory of runs and patterns and its applications: a finite Markov chain imbedding approach. World Scientific, Singapore

    Book  MATH  Google Scholar 

  • Godbole AP, Papastavridis SG, Weishaar RS (1997) Formulae and recursions for the joint distribution of success runs of several lengths. Ann Inst Stat Math 49:141–153

    Article  MathSciNet  MATH  Google Scholar 

  • Han Q, Aki S (1998) Formulae and recursions for the joint distributions of success runs of several lengths in a two-state Markov chain. Stat Probab Lett 40:203–214

    Article  MathSciNet  MATH  Google Scholar 

  • Inoue K, Aki S (2003) Generalized binomial and negative binomial distributions of order \(k\) by the \(\ell \)-overlapping enumeration scheme. Ann Inst Stat Math 55:153–167

    MathSciNet  MATH  Google Scholar 

  • Inoue K, Aki S (2005) Joint distributions of numbers of success runs of specified lengths in linear and circular sequences. Ann Inst Stat Math 57:353–368

    Article  MathSciNet  MATH  Google Scholar 

  • Inoue K, Aki S (2009) Distributions of runs and scans on higher order Markov trees. Commun Stat-Theory Methods 38(5):621–641

    Article  MathSciNet  MATH  Google Scholar 

  • Inoue K, Aki S (2010) On the conditional and unconditional distributions of the number of success runs on a circle with applications. Stat Probab Lett 80:874–885

    Article  MathSciNet  MATH  Google Scholar 

  • Inoue K, Aki S (2013) Distributions of numbers of runs and scans on directed acyclic graphs with generation. Comput Stat 28:1133–1150

    Article  MathSciNet  MATH  Google Scholar 

  • Inoue K (2015) Distributions of numbers of runs and scans on higher order Markov directed acyclic graphs with generation. Josai Math Monogr 8:3–16

    Google Scholar 

  • Jonczy J, Haenni H (2008) Network reliability evaluation with propositional directed acyclic graphs. In: Bedford T et al (eds) Advances in mathematical modeling for reliability. IOS Press, Amsterdam, pp 25–31

    Google Scholar 

  • Koutras MV, Alexandrou VA (1995) Runs, scans and urn model distributions: a unified Markov chain approach. Ann Inst Stat Math 47:743–766

    Article  MathSciNet  MATH  Google Scholar 

  • Koutras MV, Alexandrou VA (1997) Sooner waiting time problems in a sequence of trinary trials. J Appl Probab 34:593–609

    Article  MathSciNet  MATH  Google Scholar 

  • Lauritzen SL (1996) Graphical models. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Ling KD (1988) On binomial distributions of order \(k\). Stat Probab Lett 6:247–250

    Article  MathSciNet  MATH  Google Scholar 

  • Makri FS, Philippou AN (2005) On binomial and circular binomial distributions of order \(k\) for \({\ell }\)-overlapping success runs of length \(k\). Stat Pap 46:411–432

    Article  MathSciNet  MATH  Google Scholar 

  • Makri FS, Philippou AN, Psillakis ZM (2007) Success run statistics defined on an urn model. Adv Appl Probab 39:991–1019

    Article  MathSciNet  MATH  Google Scholar 

  • Mood AM (1940) The distribution theory of runs. Ann Math Stat 11:367–392

    Article  MathSciNet  MATH  Google Scholar 

  • Nakagawa T, Osaki S (1975) The discrete Weibull distribution. IEEE Trans Reliab 24(5):300–301

    Article  Google Scholar 

  • Philippou AN, Georghiou C, Philippou GN (1983) A generalized geometric distribution and some of its properties. Stat Probab Lett 1:171–175

    Article  MathSciNet  MATH  Google Scholar 

  • Rakitzis AC, Antzoulakos DL (2015) Start-up demonstration tests with three-level classification. Stat Pap 56:1–21

    Article  MathSciNet  MATH  Google Scholar 

  • Ripley BD (1996) Pattern recognition and neural networks. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Shmueli G, Cohen A (2000) Run-related probability functions applied to sampling inspection. Technometrics 42:188–202

    Article  Google Scholar 

Download references

Acknowledgments

The authors wish to thank the editor and the referees for careful reading of our paper and helpful suggestions which led to improved results.

Author information

Authors and Affiliations

  1. Faculty of Economics, Seikei University, 3-3-1 Kichijoji-Kitamachi, Musasino-shi, Tokyo, 180-8633, Japan

    Kiyoshi Inoue

  2. Department of Mathematics, Faculty of Engineering science, Kansai University, 3-3-35 Yamate-cho, Suita-shi, Osaka, 564-8680, Japan

    Sigeo Aki

Authors
  1. Kiyoshi Inoue
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sigeo Aki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kiyoshi Inoue.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Inoue, K., Aki, S. Joint distributions of numbers of runs of specified lengths on directed trees. Stat Papers 59, 249–269 (2018). https://doi.org/10.1007/s00362-016-0762-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-016-0762-y

Keywords

Mathematics Subject Classification

Search

Navigation