Abstract
In this work we are focused on multi-state systems modeled by means of a special type of semi-Markov processes. The sojourn times are seen to be independent not necessarily identically distributed random variables and assumed to belong to a general class of distributions closed under extrema that includes, in addition to some discrete distributions, several typical reliability distributions like the exponential, Weibull, and Pareto. A special parametrization is proposed for the parameters describing the system, taking thus into account various types of dependencies of the parameters on the the states of the system. We obtain maximum likelihood estimators of the parameters and plug-in type estimators are furnished for the basic quantities describing the semi-Markov system under study.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Barlow R, Wu A (1978) Coherent systems with multi-state components. Math Oper Res 3:275–281
El-Neveihi E, Proschan F, Sethuraman J (1978) Multistate coherent systems. J Appl Probab 15:675–688
Murchland J (1975) Fundamental concepts and relations for reliability analysis of multistate systems. In: Barlow R, Fussell J, Singpurwalla N (eds) Reliability and fault tree analysis: theoretical and applied aspects of system reliability. SIAM, Philadelphia, pp 581–618
Ross S (1979) Multivalued state component systems. Ann Probab 7:379–383
Block H, Savits T (1982) A decomposition of multistate monotone system. J Appl Probab 19:391–402
Hudson J, Kapur K (1982) Reliability theory for multistate systems with multistate components. Microelectron Reliab 22:1–7
Natvig B (1982) Two suggestions of how to define a multistate coherent system. Adv Appl Probab 14:434–455
El-Neveihi E, Proschan F (1984) Degradable systems: a survey of multistate system theory. Comm Stat Theory Method 13:405–432
Natvig B (1985) Multi-state coherent systems. In: Jonson N, Kotz S (eds) Encyclopedia of statistical sciences, vol 5. Wiley, New York, pp 732–735
Lisnianski A, Frenkel I, Ding Y (2010) Multi-state system reliability analysis and optimization for engineers and industrial managers. Springer, London
Lisnianski A, Levitin G (2003) Multi-state system reliability: assessment. Optimization and applications. World Scientific, Singapore
Natvig B (2011) Multistate systems reliability. Theory with applications. Wiley, New York
Limnios N, Oprisan G (2001) Semi-Markov processes and reliability. Birkhauser, Boston
Limnios N, Ouhbi B (2003) Empirical estimators of reliability and related functions for semi-Markov systems. In: Lindqvist B, Doksum K (eds) Mathematical and statistical methods in reliability 7. World Scientific, Singapore, pp 469–484
Limnios N, Ouhbi B (2006) Nonparametric estimation of some important indicators in reliability for semi-Markov processes. Stat Methodol 3:341–350
Ouhbi B, Limnios N (1996) Non-parametric estimation for semi-Markov kernels with application to reliability analysis. Appl Stoch Model Data Anal 12:209–220
Ouhbi B, Limnios N (1999) Non-parametric estimation for semi-Markov processes based on its hazard rate functions. Stat Infer Stoch Process 2(2):151–173
Balasubramanian K, Beg Bapat R (1991) On families of distributions closed under extrema. Sankhya Indian J Stat 53:375–388
Barbu V, Karagrigoriou A, Makrides A (2016) Semi-Markov modelling for multi-state systems. Method Comput Appl Probab
Barbu V, Karagrigoriou A, Makrides A (2017) Statistical inference for a general class of distributions with time varying parameters, submitted
Jiang R (2016) Weibull model with time-varying scale parameter for modeling failure processes of repairable systems, In: Frenkel I, Lisnianski A (eds) Proceedings of the second international symposium on stochastic models in reliability engineering. Life science and operations management (SMRLO16). Beer Sheva, Israel, pp 292–295, IEEE CPS, 15–18 Feb 2016, 978-1-4673-9941-8/16 (2016). doi:10.1109/SMRLO.2016.54
Yang T, (2011) Measurement of yield distribution: a time-varying distribution model. Agricultural and Applied Economics Association, Annual Meeting, 24–26 July 2011. Pittsburgh, Pennsylvania
Zhu Y, Goodwin B, Ghosh S (2010) Modeling yield risk under technological change: dynamic yield distributions and the U.S. crop insurance program. Dissertation Paper, Department of Agricultural and Resource Economics, North Carolina State University
Limnios N, Ouhbi B, Sadek A (2005) Empirical estimator of stationary distribution for semi-Markov processes. Comm Stat Theory Method 34(4):987–995
Moore H, Pyke R (1968) Estimation of the transition distribution of a Markov renewal process. Ann Inst Stat Math 20:411–424
Acknowledgements
The research work of Vlad Stefan Barbu was partially supported by the projects XTerM—Complex Systems, Territorial Intelligence and Mobility (2014–2018) and MOUSTIC—Random Models and Statistical, Informatics and Combinatorial Tools (2016–2019) within the Large Scale Research Networks from the Region of Normandy, France. The research work of Alex Karagrigoriou was completed as part of the research activities of the Laboratory of Statistics and Data Analysis of the Department of Mathematics of the University of the Aegean.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Barbu, V.S., Karagrigoriou, A. (2018). Modeling and Inference for Multi-state Systems. In: Lisnianski, A., Frenkel, I., Karagrigoriou, A. (eds) Recent Advances in Multi-state Systems Reliability. Springer Series in Reliability Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-63423-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-63423-4_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-63422-7
Online ISBN: 978-3-319-63423-4
eBook Packages: EngineeringEngineering (R0)