Several recent papers have suggested using a product estimator in Monte Carlo Markov chain sampling for estimating the volume of a convex body, the permanent of a matrix and the distribution of the first-passage time for a positive recurrent Markov chain. The present paper analyzes the properties of this estimator when each replication starts in an arbitrarily selected state. In particular, it describes a procedure for determining optimal warm-up intervals and optimal sample sizes to achieve a specified level of statistical accuracy at minimal cost. Also, it examines the variation in the optimal solution in response to changes in the parameters of the problem.
