Control-Variate Estimation Using Estimated Control Means

We study control variate estimation where the control mean itself is estimated. Control variate estimation in simulation experiments can significantly increase sampling efficiency, and has traditionally been restricted to cases where the control has a known mean. In a previous paper (Schmeiser, Taaffe, and Wang 2000), we generalized the idea of control variate estimation to the case where the control mean is only approximated. The result is a biased, but possibly useful estimator. For that case we provided a mean-square-error optimal estimator and discussed its properties. In this paper we also generalize classical control variate estimation to the case of control variates using estimated control means (CVEMs). CVEMs replace the control mean with an estimated value for the control mean, obtained from a prior simulation experiment. Although the resulting control-variate estimator is unbiased, it does introduce additional sampling error and so its properties are not the same as those of the standard control-variate estimator. We develop a CVEM estimator that minimizes the overall estimator variance. Both biased control variates (BCVs) and CVEMs can be used to improve the efficiency of stochastic simulation experiments. Their main appeal is that the restriction of having to know (deterministically) the exact value of the control mean is eliminated; thus the space of possible controls is greatly increased.