© 2002 by Biometrika Trust
On the applicability of regenerative simulation in Markov chain Monte Carlo
1 Department of Statistics, University of Florida, Gainesville, Florida 32611, U.S.Ajhobert{at}stat.ufl.edu 2 School of Statistics, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A.galin{at}stat.umn.edu 3 Department of Statistics, University of Florida, Gainesville, Florida 32611, U.S.A. presnell{at}stat.ufl.edu 4 Department of Statistics, University of Toronto, Toronto, Ontario, M5S 3G3, Canada jeff{at}math.toronto.edu
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.
Key Words: Asymptotic standard error; Burn-in; Central limit theorem; Geometric ergodicity; Minorisation condition; Slice sampler
Received October 2001. Revised April 2002