Biased control-variate estimation

We study Biased Control Variates (BCVs), whose purpose is to improve the efficiency of stochastic simulation experiments. BCVs replace the control-simulation mean with an approximation; the resulting control-variate estimator is biased. This bias may not be a significant issue for finite sample sizes, however, because our estimator minimizes the more general mean-squared-error (mse), i.e., the sum of the estimator variance plus the bias squared. After discussing an example, we review BCVs, including the mse optimal control-variate weight and associated mse performance. We then consider the relationships among bias, induced correlation, relative mse reduction, computing effort and generalized mse (gmse), assuming the use of the mse-optimal control weight, both for cases with and without bias. We define and study two estimators for the optimal control-variate weight: the Natural Estimator and the Classical Estimator. The Classical Estimator, which simply ignores the bias, can yield substantial mse reduction when the error in the approximation is small compared to the sampling error of the control-simulation.