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The continuous-time hidden Markov model based on discretization. Properties of estimators and applications

Author

Listed:
  • María Luz Gámiz

    (Universidad de Granada)

  • Nikolaos Limnios

    (Université de Technologie de Compiègne)

  • Mari Carmen Segovia-García

    (Universidad de Granada)

Abstract

In this paper we consider continuous-time hidden Markov processes (CTHMM). The model considered is a two-dimensional stochastic process $$(X_t,Y_t)$$ ( X t , Y t ) , with $$X_t$$ X t an unobserved (hidden) Markov chain defined by its generating matrix and $$Y_t$$ Y t an observed process whose distribution law depends on $$X_t$$ X t and is called the emission function. In general, we allow the process $$Y_t$$ Y t to take values in a subset of the q-dimensional real space, for some q. The coupled process $$(X_t,Y_t)$$ ( X t , Y t ) is a continuous-time Markov chain whose generator is constructed from the generating matrix of X and the emission distribution. We study the theoretical properties of this two-dimensional process using a formulation based on semi-Markov processes. Observations of the CTHMM are obtained by discretization considering two different scenarii. In the first case we consider that observations of the process Y are registered regularly in time, while in the second one, observations arrive at random. Maximum-likelihood estimators of the characteristics of the coupled process are obtained in both scenarii and the asymptotic properties of these estimators are shown, such as consistency and normality. To illustrate the model a real-data example and a simulation study are considered.

Suggested Citation

  • María Luz Gámiz & Nikolaos Limnios & Mari Carmen Segovia-García, 2023. "The continuous-time hidden Markov model based on discretization. Properties of estimators and applications," Statistical Inference for Stochastic Processes, Springer, vol. 26(3), pages 525-550, October.
    • Handle: RePEc:spr:sistpr:v:26:y:2023:i:3:d:10.1007_s11203-023-09292-0
    DOI: 10.1007/s11203-023-09292-0
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    References listed on IDEAS

    as
    1. Lin, Yiqi & Song, Xinyuan, 2022. "Order selection for regression-based hidden Markov model," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    2. Gámiz, María Luz & Limnios, Nikolaos & Segovia-García, María del Carmen, 2023. "Hidden markov models in reliability and maintenance," European Journal of Operational Research, Elsevier, vol. 304(3), pages 1242-1255.
    3. Leroux, Brian G., 1992. "Maximum-likelihood estimation for hidden Markov models," Stochastic Processes and their Applications, Elsevier, vol. 40(1), pages 127-143, February.
    4. Zhou, Jie & Song, Xinyuan & Sun, Liuquan, 2020. "Continuous time hidden Markov model for longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    5. Nikolaos Limnios, 2012. "Reliability Measures of Semi-Markov Systems with General State Space," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 895-917, December.
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    Cited by:

    1. Gámiz, M.L. & Navas-Gómez, F. & Raya-Miranda, R. & Segovia-García, M.C., 2023. "Dynamic reliability and sensitivity analysis based on HMM models with Markovian signal process," Reliability Engineering and System Safety, Elsevier, vol. 239(C).

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