IDEAS home Printed from https://ideas.repec.org/p/crs/wpaper/2004-47.html
   My bibliography  Save this paper

Regenerative Block-bootstrap for Markov Chains

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://crest.science/RePEc/wpstorage/2004-47.pdf
    File Function: Crest working paper version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Joel L. Horowitz, 2003. "Bootstrap Methods for Markov Processes," Econometrica, Econometric Society, vol. 71(4), pages 1049-1082, July.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Bertail, Patrice & Clemencon, Stephan, 2008. "Approximate regenerative-block bootstrap for Markov chains," Computational Statistics & Data Analysis, Elsevier, vol. 52(5), pages 2739-2756, January.
    2. Joel L. Horowitz, 2018. "Bootstrap methods in econometrics," CeMMAP working papers CWP53/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Wolfgang Härdle & Joel Horowitz & Jens‐Peter Kreiss, 2003. "Bootstrap Methods for Time Series," International Statistical Review, International Statistical Institute, vol. 71(2), pages 435-459, August.
    4. Joel L. Horowitz, 2018. "Bootstrap Methods in Econometrics," Papers 1809.04016, arXiv.org.
    5. Manzan, S. & Zerom, D., 2005. "A Multi-Step Forecast Density," CeNDEF Working Papers 05-05, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance.
    6. Patrice Bertail & Stéphan Clémençon, 2005. "Regeneration-based Statistics for Harris Recurrent Markov Chains," Working Papers 2005-13, Center for Research in Economics and Statistics.
    7. Manzan, Sebastiano & Zerom, Dawit, 2008. "A bootstrap-based non-parametric forecast density," International Journal of Forecasting, Elsevier, vol. 24(3), pages 535-550.
    8. Patrice Bertail & Stéphan Clémençon, 2006. "Approximate Regenerative-block Bootstrap for Markov Chains : Some Simulation Studies," Working Papers 2006-19, Center for Research in Economics and Statistics.
    9. Fernandes, Marcelo, 2006. "Financial crashes as endogenous jumps: estimation, testing and forecasting," Journal of Economic Dynamics and Control, Elsevier, vol. 30(1), pages 111-141, January.
    10. Cerqueti, Roy & Falbo, Paolo & Pelizzari, Cristian, 2017. "Relevant states and memory in Markov chain bootstrapping and simulation," European Journal of Operational Research, Elsevier, vol. 256(1), pages 163-177.
    11. Corradi, Valentina & Distaso, Walter & Fernandes, Marcelo, 2012. "International market links and volatility transmission," Journal of Econometrics, Elsevier, vol. 170(1), pages 117-141.
    12. Bitseki Penda, S. Valère, 2023. "Moderate deviation principles for kernel estimator of invariant density in bifurcating Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 282-314.
    13. Cai, Zongwu, 2003. "Nonparametric estimation equations for time series data," Statistics & Probability Letters, Elsevier, vol. 62(4), pages 379-390, May.
    14. Pan, Li & Politis, Dimitris, 2014. "Bootstrap prediction intervals for Markov processes," University of California at San Diego, Economics Working Paper Series qt7555757g, Department of Economics, UC San Diego.
    15. Sherman, Michael & Carlstein, Edward, 2004. "Confidence intervals based on estimators with unknown rates of convergence," Computational Statistics & Data Analysis, Elsevier, vol. 46(1), pages 123-139, May.
    16. Härdle, Wolfgang & Horowitz, Joel L. & Kreiss, Jens-Peter, 2001. "Bootstrap methods for time series," SFB 373 Discussion Papers 2001,59, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    17. repec:wyi:journl:002120 is not listed on IDEAS
    18. 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).
    19. Román Alejandro Mendoza Urdiales & Andrés García-Medina & José Antonio Nuñez Mora, 2021. "Measuring information flux between social media and stock prices with Transfer Entropy," PLOS ONE, Public Library of Science, vol. 16(9), pages 1-19, September.
    20. Jiyeon Lee, 2007. "First exit times for compound Poisson dams with a general release rule," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(1), pages 169-178, February.
    21. Gallefoss, Kristoffer & Hansen, Helge Hoff & Haukaas, Eirik Solli & Molnár, Peter, 2015. "What daily data can tell us about mutual funds: Evidence from Norway," Journal of Banking & Finance, Elsevier, vol. 55(C), pages 117-129.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:crs:wpaper:2004-47. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Secretariat General (email available below). General contact details of provider: https://edirc.repec.org/data/crestfr.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.