Skip to main content
SpringerLink
Log in

Estimation of the transition distributions of a Markov renewal process

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Summary

The present paper is concerned with the estimation of the transition distributions of a Markov renewal process with finitely many states. A natural estimator of the transition distributions is defined and shown to be consistent. Limiting distributions of this estimator are derived. A density for a Markov renewal process is developed to permit the definition of maximum likelihood estimators for a renewal process and for a Markov renewal process.

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

Similar content being viewed by others

References

  1. A. Albert, “Estimating the infinitesimal generator of a continuous time, finite state Markov process,”Ann. Math. Statist. 33 (1962), 727–753.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. S. Bartlett, “The frequency goodness of fit test for probability chains,”Proc. Camb. Phil. Soc., 47 (1951), 86–95.

    MATH  MathSciNet  Google Scholar 

  3. H. D. Brunk, “On the estimation of parameters restricted by inequalities,”Ann. Math. Statist., 29 (1958), 437–454.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Billingsley,Statistical inference for Markov processes, Institute of Mathematical Statistics, University of Chicago Statistical Research Monographs, University of Chicago Press, 1961.

  5. P. Billingsley, “Statistical methods in Markov chains,”Ann. Math. Statist. 32 (1961), 12–40.

    Article  MathSciNet  Google Scholar 

  6. H. Cramér,Mathematical Methods of Statistics, Princeton University Press, 1946.

  7. D. A. Darling, “The Kolmogorov-Smirnov, Cramér-von mises tests,”Ann. Math. Statist., 28 (1957), 832–838.

    Article  MathSciNet  Google Scholar 

  8. C. Derman, “Some asymptotic distribution theory for Markov chains with a denumerable number of states,”Biometrika, 43 (1956), 285–294.

    MATH  MathSciNet  Google Scholar 

  9. Ulf Grenander, “On the theory of mortality measurement, Part II,”Skand. Aktuarietidskr, 39 (1956), 129–153.

    Google Scholar 

  10. H. Hatori, “A limit theorem on (J, X)-processes,”Kodai Math. Seminar Rpts., 18 (1966), 317–321.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Keilson and D. M. G. Wishart, “A central limit theorem for processes defined on a finite Markov chain,”Proc. Camb. Phil. Soc., 60 (1964), 547–567.

    MATH  MathSciNet  Google Scholar 

  12. A. Marshall and F. Proschan, “Maximum likelihood estimation for distributions with monotone failure rate,”Ann. Math. Statist., 36 (1964), 69–77.

    Article  MathSciNet  Google Scholar 

  13. R. Pyke, “Markov renewal processes: definitions and preliminary properties,”Ann. Math. Statist., 32 (1961), 1231–1242.

    Article  MathSciNet  MATH  Google Scholar 

  14. R. Pyke and R. Schaufele, “Limit theorems for Markov renewal processes,”Ann. Math. Statist., 35 (1964), 1746–1764.

    Article  MathSciNet  MATH  Google Scholar 

  15. R. Pyke, “The weak convergence of the empirical process with random sample size,”Proc. Camb. Phil. Soc., 64 (1968), 155–160.

    Article  MATH  MathSciNet  Google Scholar 

  16. Y. Taga, “On the limiting distributions in Markov renewal processes with finitely many states,”Ann. Inst. Statist. Math., 15 (1963), 1–10.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Authors
  1. Erin H. Moore
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ronald Pyke
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The Boeing Company

About this article

Cite this article

Moore, E.H., Pyke, R. Estimation of the transition distributions of a Markov renewal process. Ann Inst Stat Math 20, 411–424 (1968). https://doi.org/10.1007/BF02911654

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02911654

Keywords

Search

Navigation