Controlling antithetic variates

Abstract

We establish and investigate a theoretical framework for controlling covariance matrices in the method of antithetic variates through control variates to further reduce estimator variance. Instead of preemptively and carefully designing an estimator vector with negatively correlated components, the proposed framework starts with a predefined estimator vector that incorporates specified control variates. The weights and control matrix are then analytically determined through matrix algebra. The joint optimality of the resulting estimator variance is ensured with respect to both the weights and the control matrix, with closed-form implementable formulas derived for the optimal parameter pair. Numerical results are provided for various typical examples to illustrate the effectiveness, potential, and challenges of the proposed framework.