The optimal linear combination of control variates in the presence of asymptotically negligible bias

The optimal linear combination of control variates is well known when the controls are assumed to be unbiased. The authors derive here the optimal linear combination of controls in the situation where asymptotically negligible bias is present. The small-sample linear control which minimizes the mean square error (MSE) is derived. When the optimal asymptotic linear control is used rather than the optimal small-sample control, the degradation in MSE is c/n³, where n is sample size and c is a known constant. This analysis is particularly relevant to the small-sample theory for control variates as applied to the steady-state estimation problem. Results for the method of multiple estimates are also given.