Batch means and spectral variance estimators in Markov chain Monte Carlo

James M. Flegal, Galin L. Jones

Research output: Contribution to journalArticlepeer-review

136 Scopus citations


Calculating a Monte Carlo standard error (MCSE) is an important step in the statistical analysis of the simulation output obtained from a Markov chain Monte Carlo experiment. An MCSE is usually based on an estimate of the variance of the asymptotic normal distribution. We consider spectral and batch means methods for estimating this variance. In particular, we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases. In addition, for the batch means and overlapping batch means methods we establish conditions ensuring consistency in the mean-square sense which in turn allows us to calculate the optimal batch size up to a constant of proportionality. Finally, we examine the empirical finite-sample properties of spectral variance and batch means estimators and provide recommendations for practitioners.

Original languageEnglish (US)
Pages (from-to)1034-1070
Number of pages37
JournalAnnals of Statistics
Issue number2
StatePublished - Apr 2010


  • Batch means
  • Markov chain
  • Monte Carlo
  • Spectral methods
  • Standard errors


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