Variance estimation and sequential stopping in steady-state simulations using linear regression

We propose a method for estimating the variance parameter of a discrete, stationary stochastic process that involves combining variance estimators at different run lengths using linear regression. We show that the estimator thus obtained is first-order unbiased and consistent under two distinct asymptotic regimes. In the first regime, the number of constituent estimators used in the regression is fixed and the numbers of observations corresponding to the component estimators grow in a proportional manner. In the second regime, the number of constituent estimators grows while the numbers of observations corresponding to each estimator remain fixed. We also show that for m-dependent stochastic processes, one can use regression to obtain asymptotically normally distributed variance estimators in the second regime. Analytical and numerical examples indicate that the new regression-based estimators give good mean-squared-error performance in steady-state simulations. The regression methodology presented in this article can also be applied to estimate the bias of variance estimators. As an example application, we present a new sequential-stopping rule that uses the estimate for bias to determine appropriate run lengths. Monte Carlo experiments indicate that this “bias-controlling” sequential-stopping method has the potential to work well in practice.