Abstract
This chapter is concerned with moment calculus and estimation of integral functionals of semi-Markov processes. We prove that the nth moment performance function verifies the Markov renewal equation and we derive a simple formula for the first two moments of the integral functional in the finite-state semi-Markov case. We propose an estimator for each one of the two quantities. then we prove their asymptotic properties. We end this chapter by proposing the confidence intervals for the first moment. As an illustration example, we give a numerical application.
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References
Asmussen, S. (1987). Applied Probability and Queues, John Wiley & Sons, New York.
Basawa, I. V. and Prakasa Rao, B. L. S. (1980). Statistical Inference for Stochastic Processes, Academic Press, London.
Ciardo, G., Marie, R. A., Sericola, B., and Trivedi, K. S. (1990). Performability analysis using semi-Markov reward processes, IEEE Transactions on Computers, 39, 1251–1264.
Csenki, A. (1995). Dependability for systems with a partitioned state space, Lecture Notes in Statistics, Vol. 90, Springer-Verlag, Berlin.
Gihman, I. I. and Skorohod, A. V. (1974). Theory of Stochastic Processes, Vol. 2, Springer-Verlag, Berlin.
Heutte, N. and Huber, C. (2002). Semi-Markov models for quality of life data with censoring, In Statistical Methods for Quality of Life Studies (Eds., M. Mesbah, B. F. Cole, and M.L.-T. Lee), pp. 207–218, Kluwer, Dordrecht.
Janssen, J. and Limnios, N. (Eds.) (1999). Semi-Markov Models and Applications, Kluwer, Dordrecht.
Korolyuk, V. S. and Limnios, N. (2005). Stochastic Systems in Merging Phase Space, World Scientific, Singapore.
Limnios, N. (2006). Estimation of the stationary distribution of semi-Markov processes with Borel state space, Statistics & Probability Letters, 76, 1536–1542.
Limnios, N. and Oprişan, G. (1999). A general framework for reliability and performability analysis of semi-Markov systems, Applied Stochastic Models in Business and Industry, 15, 353–368.
Limnios, N. and Oprişan, G. (2001). Semi–Markov Processes and Reliability, Birkhäuser, Boston.
Limnios, N. and Ouhbi, B. (2003). Empirical estimators of reliability and related functions for semi-Markov systems, In Mathematical and Statistical Methods in Reliability (Eds., B. Lindqvist and K. Doksum), World Scientific, Singapore.
Meyer, J. (1980). On evaluating the performability of degradable computer systems, IEEE Transactions on Computers, 29, 720–731.
Ouhbi, B. and Limnios, N. (1999). Non-parametric estimation for semi-Markov processes based on their hazard rate, Statist. Infer. Stoch. Processes, 2, 151–173.
Ouhbi, B. and Limnios, N. (2003). Nonparametric reliability estimation of semi-Markov processes, Journal of Statistical Planning and Inference, 109, 155–165.
Papadopoulou, A. (2004). Economic rewards in non-homogeneous semi-Markov systems, Communications in Statistics—Theory and Methods, 33, 681–696.
Parzen, E. (1999). Stochastic Processes, SIAM Classics, Philadelphia.
Prabhu, N. U. (1980). Stochastic Storage Processes, Springer-Verlag, Berlin.
Pyke, R. and Schaufele, R. (1964). Limit theorems for Markov renewal processes, Annals of Mathematical Statistics, 35, 1746–1764.
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Limnios, N., Ouhbi, B. (2008). Nonparametric Estimation of Integral Functionals for Semi-Markov Processes with Application in Reliability. In: Vonta, F., Nikulin, M., Limnios, N., Huber-Carol, C. (eds) Statistical Models and Methods for Biomedical and Technical Systems. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4619-6_28
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DOI: https://doi.org/10.1007/978-0-8176-4619-6_28
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