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On the applicability of regenerative simulation in Markov chain Monte Carlo

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  • James P. Hobert

Abstract

We consider the central limit theorem and the calculation of asymptotic standard errors for the ergodic averages constructed in Markov chain Monte Carlo. Chan & Geyer (1994) established a central limit theorem for ergodic averages by assuming that the underlying Markov chain is geometrically ergodic and that a simple moment condition is satisfied. While it is relatively straightforward to check Chan & Geyer's conditions, their theorem does not lead to a consistent and easily computed estimate of the variance of the asymptotic normal distribution. Conversely, Mykland et al. (1995) discuss the use of regeneration to establish an alternative central limit theorem with the advantage that a simple, consistent estimator of the asymptotic variance is readily available. However, their result assumes a pair of unwieldy moment conditions whose verification is difficult in practice. In this paper, we show that the conditions of Chan & Geyer's theorem are sufficient to establish the central limit theorem of Mykland et al. This result, in conjunction with other recent developments, should pave the way for more widespread use of the regenerative method in Markov chain Monte Carlo. Our results are illustrated in the context of the slice sampler. Copyright Biometrika Trust 2002, Oxford University Press.

Suggested Citation

  • James P. Hobert, 2002. "On the applicability of regenerative simulation in Markov chain Monte Carlo," Biometrika, Biometrika Trust, vol. 89(4), pages 731-743, December.
  • Handle: RePEc:oup:biomet:v:89:y:2002:i:4:p:731-743
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    Cited by:

    1. Bhattacharya, Sourabh, 2008. "Consistent estimation of the accuracy of importance sampling using regenerative simulation," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2522-2527, October.
    2. Johnson, Alicia A. & Jones, Galin L., 2015. "Geometric ergodicity of random scan Gibbs samplers for hierarchical one-way random effects models," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 325-342.
    3. Emilio De Santis & Mauro Piccioni, 2008. "Exact Simulation for Discrete Time Spin Systems and Unilateral Fields," Methodology and Computing in Applied Probability, Springer, vol. 10(1), pages 105-120, March.
    4. Pierre E. Jacob & John O’Leary & Yves F. Atchadé, 2020. "Unbiased Markov chain Monte Carlo methods with couplings," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(3), pages 543-600, July.
    5. M. A. Fleischer & S. H. Jacobson, 2002. "Scale Invariance Properties in the Simulated Annealing Algorithm," Methodology and Computing in Applied Probability, Springer, vol. 4(3), pages 219-241, September.
    6. Pierre Jacob & Christian P. Robert & Murray H. Smith, 2010. "Using Parallel Computation to Improve Independent Metropolis-Hastings Based Estimation," Working Papers 2010-44, Center for Research in Economics and Statistics.
    7. Bertail, Patrice & Clemencon, Stephan, 2008. "Approximate regenerative-block bootstrap for Markov chains," Computational Statistics & Data Analysis, Elsevier, vol. 52(5), pages 2739-2756, January.
    8. Yu Hang Jiang & Tong Liu & Zhiya Lou & Jeffrey S. Rosenthal & Shanshan Shangguan & Fei Wang & Zixuan Wu, 2022. "Markov Chain Confidence Intervals and Biases," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 11(1), pages 1-29, March.

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