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

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  • Enrique E. Alvarez

    (University of Connecticut)

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.

Suggested Citation

  • Enrique E. Alvarez, 2005. "Estimation in Stationary Markov Renewal Processes, with Application to Earthquake Forecasting in Turkey," Methodology and Computing in Applied Probability, Springer, vol. 7(1), pages 119-130, March.
    • Handle: RePEc:spr:metcap:v:7:y:2005:i:1:d:10.1007_s11009-005-6658-2
    DOI: 10.1007/s11009-005-6658-2
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    References listed on IDEAS

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    1. Brahim Ouhbi & Nikolaos Limnios, 1999. "Nonparametric Estimation for Semi-Markov Processes Based on its Hazard Rate Functions," Statistical Inference for Stochastic Processes, Springer, vol. 2(2), pages 151-173, May.
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    Cited by:

    1. Votsi, I. & Limnios, N. & Tsaklidis, G. & Papadimitriou, E., 2013. "Hidden Markov models revealing the stress field underlying the earthquake generation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(13), pages 2868-2885.
    2. Irene Votsi & Nikolaos Limnios & George Tsaklidis & Eleftheria Papadimitriou, 2012. "Estimation of the Expected Number of Earthquake Occurrences Based on Semi-Markov Models," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 685-703, September.
    3. Battarra, Maria & Balcik, Burcu & Xu, Huifu, 2018. "Disaster preparedness using risk-assessment methods from earthquake engineering," European Journal of Operational Research, Elsevier, vol. 269(2), pages 423-435.
    4. Danisman, Ozgur & Uzunoglu Kocer, Umay, 2021. "Hidden Markov models with binary dependence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 567(C).
    5. William Mohanty & Alok Mohapatra & Akhilesh Verma, 2015. "A probabilistic approach toward earthquake hazard assessment using two first-order Markov models in Northeastern India," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 75(3), pages 2399-2419, February.
    6. Vlad Stefan Barbu & Nicolas Vergne, 2019. "Reliability and Survival Analysis for Drifting Markov Models: Modeling and Estimation," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1407-1429, December.
    7. Elsa Garavaglia & Raffaella Pavani, 2011. "About Earthquake Forecasting by Markov Renewal Processes," Methodology and Computing in Applied Probability, Springer, vol. 13(1), pages 155-169, March.
    8. Yanxing Zhao & H. Nagaraja, 2011. "Fisher information in window censored renewal process data and its applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(4), pages 791-825, August.
    9. Md. Asaduzzaman & A. Latif, 2014. "A parametric Markov renewal model for predicting tropical cyclones in Bangladesh," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 73(2), pages 597-612, September.
    10. Somayajulu L. N. Dhulipala & Madeleine M. Flint, 2020. "Capabilities of multivariate Bayesian inference toward seismic hazard assessment," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 103(3), pages 3123-3144, September.

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