On the theoretical comparison of low-bias steady-state estimators

The time-average estimator is typically biased in the context of steady-state simulation, and its bias is of order 1/t, where t represents simulated time. Several "low-bias" estimators have been developed that have a lower order bias, and, to first-order, the same variance of the time-average. We argue that this kind of first-order comparison is insufficient, and that a second-order asymptotic expansion of the mean square error (MSE) of the estimators is needed. We provide such an expansion for the time-average estimator in both the Markov and regenerative settings. Additionally, we provide a full bias expansion and a second-order MSE expansion for the Meketon - Heidelberger low-bias estimator, and show that its MSE can be asymptotically higher or lower than that of the time-average depending on the problem. The situation is different in the context of parallel steady-state simulation, where a reduction in bias that leaves the first-order variance unaffected is arguably an improvement in performance.