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Some Continuous Edgeworth Expansions for Markov Chains with Applications to Bootstrap

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  • Datta, S.
  • Mccormick, W. P.

Abstract

This paper deals with the first order Edgeworth expansions for sums related to an ergodic Markov chain with general state space. In the first part of the paper, we establish certain continuity, w.r.t. the transition probability function and the initial distribution, in these expansions. In the second part, we illustrate the use of our continuous expansions in the area of bootstrap. We consider bootstrapping the distribution of the (sample) mean of a fixed real function of a Markov chain. Under a conditional non-latticeness condition, the bootstrap is shown to be second order accurate. As a second application we obtain Edgeworth expansions for the bootstrap approximation to the sampling distribution of the m.l.e. of a particular transition probability in a finite Markov chain. It is shown that the bootstrap is second order accurate and is therefore superior to the normal approximation, if the transition probability is irrational. In the other case, the exact asymptotic upper bound constant in the O(n-) rate of bootstrap approximation is determined.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:52:y:1995:i:1:p:83-106
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    Cited by:

    1. 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.
    2. Joel L. Horowitz, 2018. "Bootstrap Methods in Econometrics," Papers 1809.04016, arXiv.org.
    3. Patrice Bertail & Stéphan Clémençon, 2004. "Regenerative Block-bootstrap for Markov Chains," Working Papers 2004-47, Center for Research in Economics and Statistics.
    4. 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.
    5. 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.
    6. Joel L. Horowitz, 2018. "Bootstrap methods in econometrics," CeMMAP working papers CWP53/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

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