Skip to main content
SpringerLink
Log in

Estimation in Stationary Markov Renewal Processes, with Application to Earthquake Forecasting in Turkey

  • Original Article
  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

Consider a process in which different events occur, with random inter-occurrence times. In Markov renewal processes as well as in semi-Markov processes, the sequence of events is a Markov chain and the waiting distributions depend only on the types of the last and the next event. Suppose that the state-space is finite and that the process started far in the past, achieving stationary. Weibull distributions are proposed for the waiting times and their parameters are estimated jointly with the transition probabilities through maximum likelihood, when one or several realizations of the process are observed over finite windows. The model is illustrated with data of earthquakes of three types of severity that occurred in Turkey during the 20th century.

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.

Similar content being viewed by others

References

  • Y. Altinok and D. Kolcak, “An application of the semi-Markov model for earthquake occurrences in North Anatolia, Turkey,” Journal of the Balkan Geophysical Society vol. 2 pp. 90–99, 1999.

    Google Scholar 

  • E. E. Alvarez, Likelihood based estimation of stationary semi-Markov processes under window censoring. Ph.D. Thesis, The University of Michigan, Ann Arbor, 2003a.

  • E. E. Alvarez, “maximum likelihood estimation in alternating renewal processes under window censoring,” Technical Report 03-28, Department of Statistics, University of Connecticut, Storrs, CT, 2003b.

  • E. E. Alvarez, “Smoothed nonparametric estimation in window censored semi Markov processes,” Journal of Statistical Planning and Inference vol. 131(2) pp. 209–229, 2003c.

    Google Scholar 

  • M. Bath, “Seismic risk in Fennoscandia,” Tectono-physics vol. 57 pp. 285–295, 1979.

    Google Scholar 

  • L. A. Baxter and L. Li, “Non-parametric confidence intervals for the renewal function and the point availability,” Scandinavian Journal of Statistics vol. 21 pp. 277–287, 1994.

    Google Scholar 

  • M. Caputo, “Analysis of seismic risk, Engineering Seismology and Earthquake Engineering,” NATO Advanced Study Institutes Series, Series E: Applied Sciences vol. 3, pp. 55–86, 1974, Noordhoff-Leiden.

    Google Scholar 

  • C. A. Cornell, “Engineering seismic risk analysis,” Bulletin of the Seismological Society of America vol. 58 pp. 1583–1606, 1968.

    Google Scholar 

  • K.L. Chung, Markov Chains with Stationary Transition Probabilities, Springer: Berlin, 1967.

    Google Scholar 

  • G. E. Dinse, “A note on semi-Markov models for partially censored data,” Biometrika vol. 73(2), pp. 379–386, 1986.

    Google Scholar 

  • P. E. Greenwood and W. Wefelmeyer, “Empirical estimators for semi-Markov processes,” Mathematical Methods of Statistics vol. 5(3), pp. 299–315, 1996.

    Google Scholar 

  • S. W. Lagakos, C. J. Sommer, and M. Zelen, “Semi-Markov models for partially censored data,” Biometrika vol. 65(2), pp. 311–317, 1978.

    Google Scholar 

  • N. Limnios and G. Oprisan, Semi-Markov Processes and Reliability, Birkhauser: Boston, 2001.

    Google Scholar 

  • B. Ouhbi and N. Limnios, “Non-parametric estimation for semi-Markov processes based on its hazard rate functions,” Statistical Inference Stochastic Processes vol. 2 pp. 151–173, 1999.

    Google Scholar 

  • B. Ouhbi and N. Limnios, “Non-parametrid reliability estimation of semi-Markov processes,” Journal of Statistical Planning and Inference vol. 109 pp. 155–165, 2003.

    Google Scholar 

  • W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing, Cambridge University Press: New York, 2002.

    Google Scholar 

  • S. Ross, Applied Probability Models with Optimization Applications, Dover: New York, 1992.

    Google Scholar 

  • H. C. Shah and M. Movassate, “Seismic risk analysis of California state water project,” Proc. of the 5th European Conf. on Earthquake Engineering, 10/156, 22–25 Sept. 1975, Istanbul, 1975.

  • A. W. Van der Vaart, Asymptotic Statistics, Cambridge University Press: New York, 1998.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, University of Connecticut, 215 Glenbrook Rd, Storrs, CT, 06269-4120, USA

    Enrique E. Alvarez

Authors
  1. Enrique E. Alvarez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Enrique E. Alvarez.

Additional information

AMS 2000 Subject Classification: 60K20

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alvarez, E.E. Estimation in Stationary Markov Renewal Processes, with Application to Earthquake Forecasting in Turkey. Methodol Comput Appl Probab 7, 119–130 (2005). https://doi.org/10.1007/s11009-005-6658-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-005-6658-2

Keywords

Search

Navigation