Multiply Reflected Variance Estimators For Simulation

In a previous article, we studied a then-new class of standardized time series (STS) estimators for the asymptotic variance parameter of a stationary simulation output process. Those estimators invoke the well-known reflection principle of Brownian motion on the suitably standardized original output process to compute several "reflected" realizations of the STS, each of which is based on a single reflection point. We then calculated variance-and mean-squared-error-optimal linear combinations of the estimators formed from the singly reflected realizations. The current paper repeats the exercise except that we now examine the efficacy of employing multiple reflection points on each reflected realization of the STS. This scheme provides additional flexibility that can be exploited to produce estimators that are superior to their single-reflection-point predecessors with respect to mean-squared error. We illustrate the enhanced performance of the multiply reflected estimators via exact calculations and Monte Carlo experiments.