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Statistical inference for discretely observed Markov jump processes

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  • Mogens Bladt
  • Michael Sørensen

Abstract

Summary. Likelihood inference for discretely observed Markov jump processes with finite state space is investigated. The existence and uniqueness of the maximum likelihood estimator of the intensity matrix are investigated. This topic is closely related to the imbedding problem for Markov chains. It is demonstrated that the maximum likelihood estimator can be found either by the EM algorithm or by a Markov chain Monte Carlo procedure. When the maximum likelihood estimator does not exist, an estimator can be obtained by using a penalized likelihood function or by the Markov chain Monte Carlo procedure with a suitable prior. The methodology and its implementation are illustrated by examples and simulation studies.

Suggested Citation

  • Mogens Bladt & Michael Sørensen, 2005. "Statistical inference for discretely observed Markov jump processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(3), pages 395-410, June.
    • Handle: RePEc:bla:jorssb:v:67:y:2005:i:3:p:395-410
    DOI: 10.1111/j.1467-9868.2005.00508.x
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    Cited by:

    1. Yong Chen & Jianmin Chen, 2011. "On the Imbedding Problem for Three-State Time Homogeneous Markov Chains with Coinciding Negative Eigenvalues," Journal of Theoretical Probability, Springer, vol. 24(4), pages 928-938, December.
    2. Guglielmo D’Amico & Philippe Regnault, 2018. "Dynamic Measurement of Poverty: Modeling and Estimation," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(2), pages 305-340, November.
    3. Yasunari Inamura, 2006. "Estimating Continuous Time Transition Matrices From Discretely Observed Data," Bank of Japan Working Paper Series 06-E-7, Bank of Japan.
    4. Jia, Chen, 2016. "A solution to the reversible embedding problem for finite Markov chains," Statistics & Probability Letters, Elsevier, vol. 116(C), pages 122-130.
    5. Voß, Sebastian & Weißbach, Rafael, 2014. "A score-test on measurement errors in rating transition times," Journal of Econometrics, Elsevier, vol. 180(1), pages 16-29.
    6. Alexander Kremer & Rafael Weißbach, 2013. "Consistent estimation for discretely observed Markov jump processes with an absorbing state," Statistical Papers, Springer, vol. 54(4), pages 993-1007, November.
    7. Greig Smith & Goncalo dos Reis, 2017. "Robust and Consistent Estimation of Generators in Credit Risk," Papers 1702.08867, arXiv.org, revised Oct 2017.
    8. R. A. Hubbard & L. Y. T. Inoue & J. R. Fann, 2008. "Modeling Nonhomogeneous Markov Processes via Time Transformation," Biometrics, The International Biometric Society, vol. 64(3), pages 843-850, September.
    9. Ross, J.V. & Pagendam, D.E. & Pollett, P.K., 2009. "On parameter estimation in population models II: Multi-dimensional processes and transient dynamics," Theoretical Population Biology, Elsevier, vol. 75(2), pages 123-132.
    10. Huang, Jia-Ping & Sumita, Ushio, 2015. "Development of computational algorithms for pricing European bond options under the influence of macro-economic conditions," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 453-468.
    11. Marius Pfeuffer & Goncalo dos Reis & Greig smith, 2018. "Capturing Model Risk and Rating Momentum in the Estimation of Probabilities of Default and Credit Rating Migrations," Papers 1809.09889, arXiv.org, revised Feb 2020.
    12. Alan Riva-Palacio & Ramsés H. Mena & Stephen G. Walker, 2023. "On the estimation of partially observed continuous-time Markov chains," Computational Statistics, Springer, vol. 38(3), pages 1357-1389, September.
    13. Kremer, Alexander & Weißbach, Rafael, 2014. "Asymptotic normality for discretely observed Markov jump processes with an absorbing state," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 136-139.
    14. Yu Luo & David A. Stephens & Aman Verma & David L. Buckeridge, 2021. "Bayesian latent multi‐state modeling for nonequidistant longitudinal electronic health records," Biometrics, The International Biometric Society, vol. 77(1), pages 78-90, March.
    15. Linda Möstel & Marius Pfeuffer & Matthias Fischer, 2020. "Statistical inference for Markov chains with applications to credit risk," Computational Statistics, Springer, vol. 35(4), pages 1659-1684, December.
    16. Mogens Bladt & Michael SØrensen, 2009. "Efficient estimation of transition rates between credit ratings from observations at discrete time points," Quantitative Finance, Taylor & Francis Journals, vol. 9(2), pages 147-160.
    17. Lapshin, Viktor & Anton, Markov, 2022. "MCMC-based credit rating aggregation algorithm to tackle data insufficiency," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 68, pages 50-72.
    18. D'Amico, Guglielmo & Singh, Shakti & Selvamuthu, Dharmaraja, 2024. "Optimal investment-disinvestment choices in health-dependent variable annuity," Insurance: Mathematics and Economics, Elsevier, vol. 117(C), pages 1-15.
    19. Guglielmo D'Amico & Riccardo De Blasis & Philippe Regnault, 2020. "Confidence sets for dynamic poverty indexes," Papers 2006.06595, arXiv.org.
    20. Azam Asanjarani & Yoni Nazarathy & Peter Taylor, 2021. "A survey of parameter and state estimation in queues," Queueing Systems: Theory and Applications, Springer, vol. 97(1), pages 39-80, February.
    21. David Azriel & Paul D. Feigin & Avishai Mandelbaum, 2019. "Erlang-S: A Data-Based Model of Servers in Queueing Networks," Management Science, INFORMS, vol. 65(10), pages 4607-4635, October.
    22. Guglielmo D’Amico & Shakti Singh & Dharmaraja Selvamuthu, 2023. "Analysis of fair fee in guaranteed lifelong withdrawal and Markovian health benefits," Annals of Finance, Springer, vol. 19(3), pages 383-400, September.

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