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Reliability and Survival Analysis for Drifting Markov Models: Modeling and Estimation

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  • Vlad Stefan Barbu

    (Université de Rouen)

  • Nicolas Vergne

    (Université de Rouen)

Abstract

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.

Suggested Citation

  • Vlad Stefan Barbu & Nicolas Vergne, 2019. "Reliability and Survival Analysis for Drifting Markov Models: Modeling and Estimation," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1407-1429, December.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:4:d:10.1007_s11009-018-9682-8
    DOI: 10.1007/s11009-018-9682-8
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    References listed on IDEAS

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    1. G. Vasiliadis & G. Tsaklidis, 2008. "On the Distributions of the State Sizes of Discrete Time Homogeneous Markov Systems," Methodology and Computing in Applied Probability, Springer, vol. 10(1), pages 55-71, March.
    2. Enrique E. Alvarez, 2005. "Estimation in Stationary Markov Renewal Processes, with Application to Earthquake Forecasting in Turkey," Methodology and Computing in Applied Probability, Springer, vol. 7(1), pages 119-130, March.
    3. Brahim Ouhbi & Nikolaos Limnios, 1999. "Nonparametric Estimation for Semi-Markov Processes Based on its Hazard Rate Functions," Statistical Inference for Stochastic Processes, Springer, vol. 2(2), pages 151-173, May.
    4. Ruiz-Castro, Juan Eloy, 2016. "Markov counting and reward processes for analysing the performance of a complex system subject to random inspections," Reliability Engineering and System Safety, Elsevier, vol. 145(C), pages 155-168.
    5. Lisnianski, Anatoly & Elmakias, David & Laredo, David & Ben Haim, Hanoch, 2012. "A multi-state Markov model for a short-term reliability analysis of a power generating unit," Reliability Engineering and System Safety, Elsevier, vol. 98(1), pages 1-6.
    6. Gregory Levitin, 2005. "The Universal Generating Function in Reliability Analysis and Optimization," Springer Series in Reliability Engineering, Springer, number 978-1-84628-245-4, March.
    7. Irene Votsi & Nikolaos Limnios & George Tsaklidis & Eleftheria Papadimitriou, 2012. "Estimation of the Expected Number of Earthquake Occurrences Based on Semi-Markov Models," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 685-703, September.
    8. Vergne Nicolas, 2008. "Drifting Markov Models with Polynomial Drift and Applications to DNA Sequences," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 7(1), pages 1-45, February.
    9. Koutras, V.P. & Malefaki, S. & Platis, A.N., 2017. "Optimization of the dependability and performance measures of a generic model for multi-state deteriorating systems under maintenance," Reliability Engineering and System Safety, Elsevier, vol. 166(C), pages 73-86.
    10. E. Mathieu & Y. Foucher & P. Dellamonica & J. P. Daures, 2007. "Parametric and Non Homogeneous Semi-Markov Process for HIV Control," Methodology and Computing in Applied Probability, Springer, vol. 9(3), pages 389-397, September.
    11. Yunhui Hou & Nikolaos Limnios & Walter Schön, 2017. "On the Existence and Uniqueness of Solution of MRE and Applications," Methodology and Computing in Applied Probability, Springer, vol. 19(4), pages 1241-1250, December.
    12. Dmitrii Silvestrov & Raimondo Manca, 2017. "Reward Algorithms for Semi-Markov Processes," Methodology and Computing in Applied Probability, Springer, vol. 19(4), pages 1191-1209, December.
    13. Julien Chiquet & Nikolaos Limnios, 2006. "Estimating Stochastic Dynamical Systems Driven by a Continuous-Time Jump Markov Process," Methodology and Computing in Applied Probability, Springer, vol. 8(4), pages 431-447, December.
    14. Nikolaos Limnios, 2012. "Reliability Measures of Semi-Markov Systems with General State Space," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 895-917, December.
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    Cited by:

    1. He Yi & Lirong Cui & Narayanaswamy Balakrishnan, 2022. "On the Derivative Counting Processes of First- and Second-order Aggregated Semi-Markov Systems," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1849-1875, September.
    2. Yousif Alyousifi & Kamarulzaman Ibrahim & Mahmod Othamn & Wan Zawiah Wan Zin & Nicolas Vergne & Abdullah Al-Yaari, 2022. "Bayesian Information Criterion for Fitting the Optimum Order of Markov Chain Models: Methodology and Application to Air Pollution Data," Mathematics, MDPI, vol. 10(13), pages 1-16, June.
    3. He Yi & Lirong Cui & Narayanaswamy Balakrishnan & Jingyuan Shen, 2022. "Multi-Point and Multi-Interval Bounded-Covering Availability Measures for Aggregated Markovian Repairable Systems," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2427-2453, December.

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