Advertisement

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

  • Enrique E. AlvarezEmail author
Original Article

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.

Keywords

earthquakes marked point process Markov renewal process semi-Markov process stationarity window censoring 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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
  2. E. E. Alvarez, Likelihood based estimation of stationary semi-Markov processes under window censoring. Ph.D. Thesis, The University of Michigan, Ann Arbor, 2003a.Google Scholar
  3. 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.Google Scholar
  4. 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
  5. M. Bath, “Seismic risk in Fennoscandia,” Tectono-physics vol. 57 pp. 285–295, 1979.Google Scholar
  6. 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
  7. 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
  8. C. A. Cornell, “Engineering seismic risk analysis,” Bulletin of the Seismological Society of America vol. 58 pp. 1583–1606, 1968.Google Scholar
  9. K.L. Chung, Markov Chains with Stationary Transition Probabilities, Springer: Berlin, 1967.Google Scholar
  10. G. E. Dinse, “A note on semi-Markov models for partially censored data,” Biometrika vol. 73(2), pp. 379–386, 1986.Google Scholar
  11. 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
  12. 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
  13. N. Limnios and G. Oprisan, Semi-Markov Processes and Reliability, Birkhauser: Boston, 2001.Google Scholar
  14. 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
  15. 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
  16. 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
  17. S. Ross, Applied Probability Models with Optimization Applications, Dover: New York, 1992.Google Scholar
  18. 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.Google Scholar
  19. A. W. Van der Vaart, Asymptotic Statistics, Cambridge University Press: New York, 1998.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of ConnecticutStorrsUSA

Personalised recommendations