The Annals of Statistics

Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo

Yves F. Atchadé

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Abstract

We study the asymptotic behavior of kernel estimators of asymptotic variances (or long-run variances) for a class of adaptive Markov chains. The convergence is studied both in Lp and almost surely. The results also apply to Markov chains and improve on the existing literature by imposing weaker conditions. We illustrate the results with applications to the GARCH(1, 1) Markov model and to an adaptive MCMC algorithm for Bayesian logistic regression.

Article information

Source
Ann. Statist., Volume 39, Number 2 (2011), 990-1011.

Dates
First available in Project Euclid: 8 April 2011

Permanent link to this document
https://projecteuclid.org/euclid.aos/1302268084

Digital Object Identifier
doi:10.1214/10-AOS828

Mathematical Reviews number (MathSciNet)
MR2816345

Zentralblatt MATH identifier
1219.62125

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60C05: Combinatorial probability

Keywords
Adaptive Markov chain Monte Carlo kernel estimators of asymptotic variance

Citation

Atchadé, Yves F. Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo. Ann. Statist. 39 (2011), no. 2, 990--1011. doi:10.1214/10-AOS828. https://projecteuclid.org/euclid.aos/1302268084


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Supplemental materials

  • Supplementary material: Supplement to “Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo”. The proofs of Theorems 4.1–4.3 require some technical and lengthy arguments that we develop in this supplement.