Risk-Efficient Sequential Simulation Estimators

Using steady state mean estimation as the prototypical context, we present a decision-theoretic framework for sequentially estimating quantities associated with an observable discrete-time stochastic process. Our framework includes weights for estimator quality and a linear cost of sampling. We first show that the optimal time to stop sampling in the hypothetical case when the autocovariance function of the process is known is the square root of the relative cost and the area under the autocovariance function. This expression inspires a sequential procedure that uses a partially overlapping batch means estimator to stand-in for the area under the autocovariance function. The sequential procedure is asymptotically optimal in the sense that the ratio of its risk and that of the optimal risk in the hypothetical scenario approaches unity in a certain asymptotic regime. The nature of our analysis hints at a general optimality principle that may be more generally prevalent.