Reliability and Survival Analysis for Drifting Markov Models: Modeling and Estimation
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In this work we focus on multi-state systems modeled by means of a particular class of non-homogeneous Markov processes introduced in Vergne (Stat Appl Genet Mol Biol 7(1):1–45, 2008), called drifting Markov processes. The main idea behind this type of processes is to consider a non-homogeneity that is “smooth”, of a known shape. More precisely, the Markov transition matrix is assumed to be a linear (polynomial) function of two (several) Markov transition matrices. For this class of systems, we first obtain explicit expressions for reliability/survival indicators of drifting Markov models, like reliability, availability, maintainability and failure rates. Then, under different statistical settings, we estimate the parameters of the model, obtain plug-in estimators of the associated reliability/survival indicators and investigate the consistency of the estimators. The quality of the proposed estimators and the model validation is illustrated through numerical experiments.
KeywordsDrifting Markov chains Multi-state systems Reliability theory Survival analysis Estimation Asymptotic properties
Mathematics Subject Classification (2010)60J10 60K15 90B25 62N02 62F12
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This research work was partially supported by the projects MMATFAS - Mathematical Methods for System Reliability and Survival Analysis (2014–2015) and MOUSTIC–Random Models and Statistical, Informatics and Combinatorics Tools (2016–2019), within the Large Scale Research Networks from the Region of Normandy, France.
The authors would like to thank their colleagues Dr. Anatoly Lisnianski and Dr. Gregory Levitin from the Israel Electric Corporation Ltd. for pointing out important aspects and references on multi-state systems and also to Prof. Ilia Frenkel and his colleagues from Sami Shamoon College of Engineering, Israel, for organizing the series of conferences International Symposium on Stochastic Models in Reliability Engineering, Life Science and Operations Management, to which, indirectly, the research developed in this article owes much.
- Barbu VS, Limnios N (2004) Discrete time semi-Markov processes for reliability and survival analysis–a nonparametric estimation approach. In: Balakrishnan N, Nikulin M, Mesbah M, Limnios N (eds) Parametric and semiparametric models with applications to reliability, survival analysis and quality of life, Collection Statistics for Industry and Technology. Birkhäuser, Boston, pp 487–502Google Scholar
- Barbu VS, Limnios N (2006) Nonparametric estimation for failure rate functions of discrete time semi-Markov processes. In: Nikulin M, Commenges D, Huber-Carol C (eds) Probability, statistics and modelling in public health. Springer, pp 53–72Google Scholar
- Barbu VS, Limnios N (2008a) Reliability of semi-Markov systems in discrete time: modeling and estimation. In: Misra KB (ed) Handbook on performability engineering. Springer, pp 369–380Google Scholar
- Barbu VS, Limnios N (2010) Some algebraic methods in semi-Markov processes. In: Viana MAG, Wynn HP (eds) Algebraic methods in statistics and probability, vol 2, series contemporary mathematics edited by AMS. Urbana, pp 19-35Google Scholar
- Levitin G (2005) The universal generating function in reliability analysis and optimization. Springer, LondonGoogle Scholar