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Analysis of MCMC algorithms for Bayesian linear regression with Laplace errors

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  • Choi, Hee Min
  • Hobert, James P.

Abstract

Let π denote the intractable posterior density that results when the standard default prior is placed on the parameters in a linear regression model with iid Laplace errors. We analyze the Markov chains underlying two different Markov chain Monte Carlo algorithms for exploring π. In particular, it is shown that the Markov operators associated with the data augmentation (DA) algorithm and a sandwich variant are both trace-class. Consequently, both Markov chains are geometrically ergodic. It is also established that for each i∈{1,2,3,…}, the ith largest eigenvalue of the sandwich operator is less than or equal to the corresponding eigenvalue of the DA operator. It follows that the sandwich algorithm converges at least as fast as the DA algorithm.

Suggested Citation

  • Choi, Hee Min & Hobert, James P., 2013. "Analysis of MCMC algorithms for Bayesian linear regression with Laplace errors," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 32-40.
    • Handle: RePEc:eee:jmvana:v:117:y:2013:i:c:p:32-40
    DOI: 10.1016/j.jmva.2013.02.004
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    References listed on IDEAS

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    1. Khare, Kshitij & Hobert, James P., 2012. "Geometric ergodicity of the Gibbs sampler for Bayesian quantile regression," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 108-116.
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    4. Roy, Vivekananda & Hobert, James P., 2010. "On Monte Carlo methods for Bayesian multivariate regression models with heavy-tailed errors," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1190-1202, May.
    5. Hideo Kozumi & Genya Kobayashi, 2009. "Gibbs Sampling Methods for Bayesian Quantile Regression," Discussion Papers 2009-02, Kobe University, Graduate School of Business Administration.
    6. Yu, Keming & Moyeed, Rana A., 2001. "Bayesian quantile regression," Statistics & Probability Letters, Elsevier, vol. 54(4), pages 437-447, October.
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    Cited by:

    1. Jung, Yeun Ji & Hobert, James P., 2014. "Spectral properties of MCMC algorithms for Bayesian linear regression with generalized hyperbolic errors," Statistics & Probability Letters, Elsevier, vol. 95(C), pages 92-100.
    2. James P. Hobert & Kshitij Khare, 2016. "Discussion," International Statistical Review, International Statistical Institute, vol. 84(3), pages 349-356, December.
    3. Zeng, Zijian & Li, Meng, 2021. "Bayesian median autoregression for robust time series forecasting," International Journal of Forecasting, Elsevier, vol. 37(2), pages 1000-1010.
    4. Yunwen Yang & Huixia Judy Wang & Xuming He, 2016. "Posterior Inference in Bayesian Quantile Regression with Asymmetric Laplace Likelihood," International Statistical Review, International Statistical Institute, vol. 84(3), pages 327-344, December.
    5. Chamberlain Mbah & Kris Peremans & Stefan Van Aelst & Dries F. Benoit, 2019. "Robust Bayesian seemingly unrelated regression model," Computational Statistics, Springer, vol. 34(3), pages 1135-1157, September.
    6. Zijian Zeng & Meng Li, 2020. "Bayesian Median Autoregression for Robust Time Series Forecasting," Papers 2001.01116, arXiv.org, revised Dec 2020.

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