On the mean-squared error of variance estimators for computer simulations

Given an output process generated by a steady-state simulation, we give expressions for the mean-squared error (MSE) of several well-known estimators of the associated variance parameter. The variance estimators are based on the method of nonoverlapping batch means and on the method of standardized time series applied to overlapping batch means. Under certain conditions, the resulting expressions are used to minimize the MSE with respect to the batch size, where the optimal batch size is expressed as a function of the simulation run length and certain moment properties of the output process. The ultimate objective is to exploit these results to construct new variance estimators with improved accuracy and efficiency, and to provide useful guidelines on setting the batch size in practice.