Conditional failure occurrence rates for semi-Markov chainsFootnote^*

* An earlier version of the paper is available as an open access archive paper in the link https://hal.archives-ouvertes.fr/hal-01761067/document.

References

• V. Barbu, J. Bulla, and A. Maruotti, Estimation of the stationary distribution of a semi-Markov chain, Reliab. Stat. Stud. 5 (2012), pp. 15–26.
   Google Scholar
• V. Barbu, A. Karagrigoriou, and A. Makrides, Semi-Markov modelling for multi-state systems, Methodol. Comput. Appl. Probab. 19 (2017), pp. 1011–1028. doi: 10.1007/s11009-016-9510-y
   Web of Science ®Google Scholar
• V. Barbu and N. Limnios, Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications: Their Use in Reliability and DNA Analysis, Lecture Notes in Statistics, Springer, New York, 2008.
   Google Scholar
• V.S. Barbu, G. D'Amico, R. Manca, and P. Petroni, Step semi-Markov models and application to manpower management, ESAIM: Probab. Stat. 20 (2016), pp. 555–571. doi: 10.1051/ps/2016025
   Web of Science ®Google Scholar
• L.E. Baum and T. Petrie, Statistical inference for probabilistic functions of finite state Markov chains, Ann. Math. Statist. 37 (1966), pp. 1554–1563. doi: 10.1214/aoms/1177699147
   Google Scholar
• O. Chryssaphinou, M. Karaliopoulou, and N. Limnios, On discrete time semi-Markov chains and applications in word occurrences, Commun. Stat. Theory Methods 37 (2008), pp. 1306–1322. doi: 10.1080/03610920701713328
   Web of Science ®Google Scholar
• G. D'Amico, Rate of occurrence of failures (ROCOF) of higher-order for Markov processes: Analysis, inference and application to financial credit ratings, Methodol. Comput. Appl. Probab. 17 (2015), pp. 929–949. doi: 10.1007/s11009-015-9437-8
   Web of Science ®Google Scholar
• G. D'Amico, F. Petroni, and F. Prattico, Performance analysis of second order semi-Markov chains: an application to wind energy production, Methodol. Comput. Appl. Probab. 17 (2015), pp. 781–794. doi: 10.1007/s11009-013-9394-z
   Web of Science ®Google Scholar
• S. Georgiadis, First hitting probabilities for semi-Markov chains and estimation, Commun. Stat. Theory Method 46 (2017), pp. 2435–2446. doi: 10.1080/03610926.2015.1045080
   Web of Science ®Google Scholar
• S. Georgiadis, N. Limnios, and I. Votsi, Reliability and probability of first occurred failure for discrete-time semi-Markov systems, in Applied Reliability Engineering and Risk Analysis: Probabilistic Models and Statistical Inference. I. Frenkel, A. Karagrigoriou, A. Lisnianski, and A. Kleyner, eds., Wiley & Sons, New York, 2013, pp. 167–179.
   Google Scholar
• R. Howard, Dynamic Probabilistic Systems, Vol. II, Wiley, New York, 1971.
   Google Scholar
• N. Limnios, Reliability measures of semi-Markov systems with general state space, Methodol. Comput. Appl. Probab. 14 (2012), pp. 895–917. doi: 10.1007/s11009-011-9211-5
   Web of Science ®Google Scholar
• N. Limnios and G. Oprişan, Convergence of Probability Measures, 2nd ed., Wiley, New York, 2001.
   Google Scholar
• N. Limnios and B. Ouhbi, Nonparametric estimation of some important indicators in reliability for semi-Markov processes, Stat. Methodol. 3 (2006), pp. 341–350. doi: 10.1016/j.stamet.2005.10.003
   Google Scholar
• C.J. Mode and C.K. Sleeman, Stochastic Processes in Epidemiology: HIV/AIDS, Other Infectious Diseases, and Computers, World Scientific, Singapore, 2000.
   Google Scholar
• B. Ouhbi and N. Limnios, The rate of occurrence of failures for semi-Markov processes and estimation, Stat. Probab. Lett. 59 (2002), pp. 245–255. doi: 10.1016/S0167-7152(02)00139-6
   Web of Science ®Google Scholar
• A. Sadek and N. Limnios, Asymptotic properties for maximum likelihood estimators for reliability and failure rates of Markov chains, Commun. Stat. Theory Method 31 (2002), pp. 1837–1861. doi: 10.1081/STA-120014916
   Web of Science ®Google Scholar
• S. Trevezas and N. Limnios, Exact MLE and asymptotic properties for nonparametric semi-Markov models, J. Nonparametr. Stat. 23 (2011), pp. 719–739. doi: 10.1080/10485252.2011.555543
   Web of Science ®Google Scholar
• A.W. van der Vaart, Asymptotic Statistics, Cambridge University Press, New York, 1998.
   Google Scholar
• P.-C.G. Vassiliou and A.A. Papadopoulou, Asymptotic behavior of nonhomogeneous semi-Markov systems, Linear Algebra Appl. 210 (1992), pp. 153–198.
   Web of Science ®Google Scholar
• P.-C.G. Vassiliou and A.A. Papadopoulou, Non-homogeneous semi-Markov systems and maintainability of the state sizes, J. Appl. Probab. 29 (1992), pp. 519–534. doi: 10.2307/3214890
   Web of Science ®Google Scholar
• I. Votsi, Reliability indicators for hidden Markov renewal models, in Reliability Engineering: Theory and Applications. I. Vonta and M. Ram, eds., CRC Press, Boca Raton, 2019, pp. 105–120.
   Google Scholar
• I. Votsi and A. Brouste, Confidence intervals for risk indicators in semi-Markov models: An application to wind energy production, J. Appl. Stat. (2019). doi:10.1080/02664763.2019.1566449.
   Google Scholar
• I. Votsi and N. Limnios, Estimation of the intensity of the hitting time for semi-Markov chains and hidden Markov renewal chains, J. Nonparametr. Stat. 27 (2015), pp. 149–166. doi: 10.1080/10485252.2015.1009369
   Web of Science ®Google Scholar
• I. Votsi, N. Limnios, G. Tsaklidis, and E. Papadimitriou, Estimation of the expected number of earthquake occurrences based on semi-Markov models, Methodol. Comput. Appl. Probab. 14 (2012), pp. 685–703. doi: 10.1007/s11009-011-9257-4
   Web of Science ®Google Scholar
• I. Votsi, N. Limnios, G. Tsaklidis, and E. Papadimitriou, Hidden semi-Markov modeling for the estimation of earthquake occurrence rates, Commun. Stat. Theory Methods 43 (2014), pp. 1484–1502. doi: 10.1080/03610926.2013.857414
   Web of Science ®Google Scholar
• C. Xu, M.-Z. Wu, and M. Hamdaoui, Mixed integer multi-objective optimization of composite structures with frequency-dependent interleaved viscoelastic damping layers, Comput. Struct. 172 (2016), pp. 81–92. doi: 10.1016/j.compstruc.2016.05.006
   Web of Science ®Google Scholar
• L. Yeh, The rate of occurrence of failures, J. Appl. Probab. 34 (1997), pp. 234–247. doi: 10.2307/3215190
   Web of Science ®Google Scholar