Estimation of the intensity of the hitting time for semi-Markov chains and hidden Markov renewal chains

References

• Anselone P.M. (1960), ‘Ergodic Theory for Discrete Semi-Markov Chains’, Duke Mathematical Journal, 27, 33–40. doi: 10.1215/S0012-7094-60-02703-4
   Web of Science ®Google Scholar
• Barbu V., and Limnios N. ( 2006), ‘Maximum Likelihood Estimation for Hidden Semi-Markov Models’, Comptes Rendus de l'Académie des Sciences Paris, 342, 201–205.
   Web of Science ®Google Scholar
• Barbu V., and Limnios N. ( 2008), Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications, Lecture Notes in Statistics, New York: Springer.
   Google Scholar
• Barbu V., Bulla J., and Maruotti A. ( 2012), ‘Estimation of the Stationary Distribution of a Semi-Markov Chain’, Journal of Reliability and Statistical Studies, 5, 15–26.
   Google Scholar
• Baum L.E., and Petrie T. ( 1966), ‘Statistical Inference for Probabilistic Functions of Finite State Markov Chains’, The Annals of Mathematical Statistics, 37, 1554–1563. doi: 10.1214/aoms/1177699147
   Google Scholar
• Bickel P.J., Ritov Y., and Rydén T. ( 1998), ‘Asymptotic Normality of the Maximum-Likelihood Estimator for General Hidden Markov Models’, The Annals of Statistics, 26(4), 1614–1635. doi: 10.1214/aos/1024691255
   Web of Science ®Google Scholar
• Billingsley P. (1961), ‘Statistical Methods in Markov Chains’, The Annals of Mathematical Statistics, 32(1):12–40. doi: 10.1214/aoms/1177705136
   Google Scholar
• Billingsley P. (1968), Convergence of Probability Measures, New York: John Wiley.
   Google Scholar
• Chryssaphinou O., Karaliopoulou M., and Limnios N. ( 2008), ‘On Discrete Time Semi-Markov Chains and Applications in Words Occurrences’, Communications in Statistics – Theory and Methods, 37, 1306–1322. doi: 10.1080/03610920701713328
   Web of Science ®Google Scholar
• Douc R., and Matias C. ( 2001), ‘Asymptotics of the Maximum Likelihood Estimator for General Hidden Markov Models’, Bernoulli, 7(3):381–420. doi: 10.2307/3318493
   Web of Science ®Google Scholar
• Ferguson J. (1980), ‘Variable Duration Models for Speech’, in Proceedings of the Symposium on the Application of Hidden Markov Models to Text and Speech, Princeton, NJ, pp. 143–179.
   Google Scholar
• Howard R. (1971), Dynamic Probabilistic Systems, New York: Wiley.
   Google Scholar
• Iosifescu M., Limnios N., and Oprişan G. ( 2010), Introduction to Stochastic Models (1st edition). London: John Wiley.
   Google Scholar
• Limnios N. (2012), ‘Reliability Measures of Semi-Markov Systems with General State Space’, Methodology and Computing in Applied Probability, 14(4):895–917. doi: 10.1007/s11009-011-9211-5
   Web of Science ®Google Scholar
• Limnios N., and Oprişan G. ( 2001), Semi-Markov Processes and Reliability, Boston, MA: Birkhäuser.
   Google Scholar
• Malefaki S., Trevezas S., and Limnios N. ( 2009), ‘An EM and a Stochastic Version of the EM Algorithm for Nonparametric Hidden Semi-Markov Models’, Communications in Statistics – Simulation and Computation, 14(4):895–917.
   Google Scholar
• Mode C.J., and Sleeman C.K. ( 2000), Stochastic Processes in Epidemiology: HIV/AIDS, Other Infectious Diseases, and Computers, Singapore; World Scientific.
   Google Scholar
• Ouhbi B., and Limnios N. ( 2002), ‘The Rate of Occurrence of Failures for Semi-Markov Processes and Estimation’, Statistics & Probability Letters, 59, 245–255. doi: 10.1016/S0167-7152(02)00139-6
   Web of Science ®Google Scholar
• Rydén T. (1995), ‘Consistent and Asymptotically Normal Parameter Estimates for Markov Modulated Poisson Process’, Scandinavian Journal of Statistics, 22(3), 295–303.
   Web of Science ®Google Scholar
• Trevezas S., and Limnios N. ( 2009), ‘Maximum Likelihood Estimation for General Hidden Semi-Markov Processes with Backward Recurrence Time Dependence’, Journal of Mathematical Sciences (New York), 163(3), 262–274. doi: 10.1007/s10958-009-9675-9
   Google Scholar
• van der Vaart A.W. (1998), Asymptotic Statistics, Cambridge: Cambridge University Press.
   Google Scholar
• Vassiliou P.-C.G., and Papadopoulou A.A. ( 1992), ‘Non-Homogeneous Semi-Markov Systems and Maintainability of the State Sizes’, Journal of Applied Probability, 29, 519–534. doi: 10.2307/3214890
   Web of Science ®Google Scholar
• Vassiliou P.-C.G., and Papadopoulou A.A. ( 1994), ‘Asymptotic Behavior of NonHomogeneous Semi-Markov Systems’, Linear Algebra Application, 210, 153–198. doi: 10.1016/0024-3795(94)90470-7
   Web of Science ®Google Scholar
• Votsi I., Limnios N., Tsaklidis G., and Papadimitriou E. ( 2012), ‘Estimation of the Expected Number of Earthquake Occurrences Based on Semi-Markov Models’, Methodology and Computing in Applied Probability, 14(3), 685–703. doi: 10.1007/s11009-011-9257-4
   Web of Science ®Google Scholar
• Votsi I., Limnios N., Tsaklidis G., and Papadimitriou E. ( 2014), ‘Hidden Semi-Markov Modeling for the Estimation of Earthquake Occurrence Rates’, Communications in Statistics – Theory and Methods, 43, 1484–1502. doi: 10.1080/03610926.2013.857414
   Web of Science ®Google Scholar
• Yeh L. (1997), ‘The Rate of Occurrence of Failures’, Journal of Applied Probability, 34, 234–247. doi: 10.2307/3215190
   Web of Science ®Google Scholar