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Regenerative Block-bootstrap for Markov Chains

Author

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  • Patrice Bertail

    (Crest)

  • Stéphan Clémençon

    (Crest)

Abstract

This paper introduces a specific Bootstrap method for positiverecurrent Markov chains, based on the regenerative method and the Nummelinsplitting technique. The main idea underlying this construction consists in generatinga sequence of approximate pseudo-renewal times for a Harris chain X fromdata X1; :::; Xn and the parameters of a minorization condition satisfied by itstransition probability kernel and then applying a variant of the methodology proposedby Datta &McCormick (1993) for bootstrapping additive functionals of typen¡1Pni=1 f(Xi)when the chain possesses an atom. We prove that, in the atomiccase, our method inherits the accuracy of the Bootstrap in the i.i.d. case up toOP(n¡1) under weak conditions. In the general (non necessarily) stationary case,asymptotic validity for this resampling procedure is established, provided that aconsistent estimator of the transition kernel may be computed. The second ordervalidity (up to a rate close to OP (n¡1)for regular stationary Markov Chains) isobtained in the stationary case. Applications to specific Markovian models arediscussed, together with some simulation results.

Suggested Citation

  • Patrice Bertail & Stéphan Clémençon, 2004. "Regenerative Block-bootstrap for Markov Chains," Working Papers 2004-47, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2004-47
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    References listed on IDEAS

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    1. M. Rajarshi, 1990. "Bootstrap in Markov-sequences based on estimates of transition density," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(2), pages 253-268, June.
    2. George Roussas, 1969. "Nonparametric estimation in Markov processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 21(1), pages 73-87, December.
    3. J. Michael Harrison & Sidney I. Resnick, 1976. "The Stationary Distribution and First Exit Probabilities of a Storage Process with General Release Rule," Mathematics of Operations Research, INFORMS, vol. 1(4), pages 347-358, November.
    4. Datta, S. & Mccormick, W. P., 1995. "Some Continuous Edgeworth Expansions for Markov Chains with Applications to Bootstrap," Journal of Multivariate Analysis, Elsevier, vol. 52(1), pages 83-106, January.
    5. Joel L. Horowitz, 2003. "Bootstrap Methods for Markov Processes," Econometrica, Econometric Society, vol. 71(4), pages 1049-1082, July.
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