Control-variate estimation using estimated control means

This article studies 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 the current authors 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, a mean square error optimal estimator was provided and its properties were discussed. This article generalizes classical control variate estimation to the case of Control Variates using Estimated 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. A CVEM estimator is developed that minimizes the overall estimator variance. Both biased control variates 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.