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\begin{center}{\sc Regeneration in MCMC}\end{center} 

We consider the implementation of regenerative simulation (RS) in
Markov chain Monte Carlo.  RS has been used successfully to calculate
rigorous Monte Carlo standard errors and to construct adaptive MCMC
methods when a full-dimensional update is used and the quantity
being estimated is an expectation.  We extend this work to the case
where a quantile is estimated.  Unfortunately, RS is not generally
practical in MCMC, especially in high-dimensional settings with a
full-dimensional update.  Thus we also consider implementation of RS
in the situation where we update a high-dimensional chain one variable
(or sub-block of variables) at a time, so-called component-wise
MCMC. However, implementation of RS in component-wise settings
presents additional difficulties (both theoretical and practical)
which are sometimes insurmountable.  We derive conditions which permit
theoretically valid use of RS in component-wise MCMC settings and
consider when the use of RS is inappropriate.  We then use these
conditions to construct adaptive component-wise Metropolis-Hastings
samplers as well as a method for calculating Monte Carlo standard
errors for both adaptive and non-adaptive settings.  Our results are
illustrated in several examples.

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