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On a multidimensional general bootstrap for empirical estimator of continuous-time semi-Markov kernels with applications

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  • Salim Bouzebda
  • Chrysanthi Papamichail
  • Nikolaos Limnios

Abstract

The present paper introduces a general notion and presents results of bootstrapped empirical estimators of the semi-Markov kernels and of the conditional transition distributions for semi-Markov processes with countable state space, constructed by exchangeably weighting the sample. Our proposal provides a unification of bootstrap methods in the semi-Markov setting including, in particular, Efron's bootstrap. Asymptotic properties of these generalised bootstrapped empirical distributions are obtained, under mild conditions by a martingale approach. We also obtain some new results on the weak convergence of the empirical semi-Markov processes. We apply these general results in several statistical problems such as the construction of confidence bands and the goodness-of-fit tests where the limiting distributions are derived under the null hypothesis. Finally, we introduce the quantile estimators and their bootstrapped versions in the semi-Markov framework and we establish their limiting laws by using the functional delta methods. Our theoretical results and numerical examples by simulations demonstrate the merits of the proposed techniques.

Suggested Citation

  • Salim Bouzebda & Chrysanthi Papamichail & Nikolaos Limnios, 2018. "On a multidimensional general bootstrap for empirical estimator of continuous-time semi-Markov kernels with applications," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 30(1), pages 49-86, January.
  • Handle: RePEc:taf:gnstxx:v:30:y:2018:i:1:p:49-86
    DOI: 10.1080/10485252.2017.1404059
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Malefaki, Sonia & Limnios, Nikolaos & Dersin, Pierre, 2014. "Reliability of maintained systems under a semi-Markov setting," Reliability Engineering and System Safety, Elsevier, vol. 131(C), pages 282-290.
    4. B. Ouhbi & N. Limnios, 1996. "Non‐parametric estimation for semi‐Markov kernels with application to reliability analysis," Applied Stochastic Models and Data Analysis, John Wiley & Sons, vol. 12(4), pages 209-220, December.
    5. 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.
    6. Salim Bouzebda & Mohamed Cherfi, 2012. "General Bootstrap for Dual ϕ-Divergence Estimates," Journal of Probability and Statistics, Hindawi, vol. 2012, pages 1-33, February.
    7. 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.
    8. 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.
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