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Bootstrap of Reliability Indicators for Semi-Markov Processes

Author

Listed:
  • Irene Votsi

    (Le Mans Université
    Université de Lorraine)

  • Salim Bouzebda

    (Université de Technologie de Compiègne)

Abstract

The present paper concerns bootstrap and kernel-type estimators of some functionals of the semi-Markov kernel (SMK) coming from the area of reliability. This type of functionals could be found in real-world applications encountered in different scientific fields. First, the weak convergence of the bootstrap kernel-type estimators of the SMK and the conditional sojourn time distributions are proved. The convergence in probability and the asymptotic normality of their derivatives are obtained. Second, new estimators of reliability indicators such as the mean time to failure, the reliability and the availability are introduced. These estimators are based on exchangeable weighted bootstrap approaches, kernel-type approaches or both of them. The method of bootstrap is employed since it is widely used to solve problems in statistics related to limiting distributions. The bootstrap estimators are shown to be asymptotically normal. The asymptotic properties of the estimators are illustrated by means of Monte-Carlo simulations.

Suggested Citation

  • Irene Votsi & Salim Bouzebda, 2025. "Bootstrap of Reliability Indicators for Semi-Markov Processes," Methodology and Computing in Applied Probability, Springer, vol. 27(1), pages 1-18, March.
    • Handle: RePEc:spr:metcap:v:27:y:2025:i:1:d:10.1007_s11009-024-10125-7
    DOI: 10.1007/s11009-024-10125-7
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    References listed on IDEAS

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    1. 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.
    2. Bouzebda, Salim & Limnios, Nikolaos, 2013. "On general bootstrap of empirical estimator of a semi-Markov kernel with applications," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 52-62.
    3. S. Georgiadis & N. Limnios, 2012. "A multidimensional functional central limit theorem for an empirical estimator of a continuous-time semi-Markov kernel," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(4), pages 1007-1017, December.
    4. E. del Barrio & C. Matrán, 2000. "The Weighted Bootstrap Mean for Heavy-Tailed Distributions," Journal of Theoretical Probability, Springer, vol. 13(2), pages 547-569, April.
    5. Salim Bouzebda & Mohamed Chaouch & Sultana Didi Biha, 2022. "Asymptotics for function derivatives estimators based on stationary and ergodic discrete time processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(4), pages 737-771, August.
    6. I. Votsi & N. Limnios, 2015. "Estimation of the intensity of the hitting time for semi-Markov chains and hidden Markov renewal chains," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(2), pages 149-166, June.
    7. Ronald W. Butler & Douglas A. Bronson, 2012. "Bootstrap confidence bands for sojourn distributions in multistate semi-Markov models with right censoring," Biometrika, Biometrika Trust, vol. 99(4), pages 959-972.
    8. Soukarieh, Inass & Bouzebda, Salim, 2023. "Renewal type bootstrap for increasing degree U-process of a Markov chain," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    9. Dragan Radulović, 2004. "Renewal type bootstrap for Markov chains," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(1), pages 147-192, June.
    10. Inass Soukarieh & Salim Bouzebda, 2022. "Exchangeably Weighted Bootstraps of General Markov U -Process," Mathematics, MDPI, vol. 10(20), pages 1-42, October.
    11. Arnold Janssen, 2005. "Resampling student'st-type statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(3), pages 507-529, September.
    12. Salim Bouzebda & Nikolaos Limnios, 2022. "The uniform CLT for the empirical estimator of countable state space semi-Markov kernels indexed by functions with applications," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 34(4), pages 758-788, October.
    13. Salim Bouzebda & Boutheina Nemouchi, 2023. "Weak-convergence of empirical conditional processes and conditional U-processes involving functional mixing data," Statistical Inference for Stochastic Processes, Springer, vol. 26(1), pages 33-88, April.
    14. Guang Cheng, 2015. "Moment Consistency of the Exchangeably Weighted Bootstrap for Semiparametric M-estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(3), pages 665-684, September.
    15. Irene Votsi, 2019. "Conditional failure occurrence rates for semi-Markov chains," Journal of Applied Statistics, Taylor & Francis Journals, vol. 46(15), pages 2722-2743, November.
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