Abstract
For a continuous-time Markov process, commonly, only discrete-time observations are available. We analyze multiple observations of a homogeneous Markov jump process with an absorbing state. We establish consistency of the maximum likelihood estimator, as the number of Markov processes increases. To accomplish uniform convergence in the continuous mapping theorem, we use the continuity of the transition probability in the parameters, the compactness of the parameter space and the boundedness of probabilities. We allow for a stochastic time-grid of observation points with different intensities for each observation process. Furthermore, we account for right censoring. The estimate is obtained via the EM algorithm with an E-step given in closed form. In our empirical application of credit rating histories, we fit the model of Weißbach and Mollenhauer (J Korean Stat Soc 40:469–485, 2011) and find marked differences, compared to the continuous-time analysis.
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Notes
Discussion at the Sixth Autumn Symposium of the Research Training Group “Statistical Modelling”, November 20th and 21st, 2009, Bommerholz, Germany.
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Acknowledgments
We would like to thank Friedrich Liese for his valuable suggestions. The financial support from the Deutsche Forschungsgemeinschaft is also gratefully acknowledged (Grant WE3573/2). All computations have been processed in MATLAB.
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Kremer, A., Weißbach, R. Consistent estimation for discretely observed Markov jump processes with an absorbing state. Stat Papers 54, 993–1007 (2013). https://doi.org/10.1007/s00362-013-0515-0
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DOI: https://doi.org/10.1007/s00362-013-0515-0