Weighted Batch Means for Confidence Intervals in Steady-State Simulations

We propose a new procedure for providing confidence-interval estimators of the mean of a covariance-stationary process. The procedure, a modification of the method of batch means, is an improvement over existing methods when the process displays strong correlation and a comparatively small number of observations is available.

We assign weights to the observations within a batch. The weights are determined by the order of the time-series model fit to the process and by its estimated parameters. For a given model and its parameters, the weights minimize the variance of the weighted point estimator of the mean; the point and variance estimators formed when these optimal weights are applied are unbiased. The time-series identification procedure and estimation of the parameters and weights bring in bias. Formulas for optimal weights are given for AR(p) and MA(1) processes.

Experiments were conducted on AR(1), AR(2), and MA(2) processes as well as on the M/M/1 queue delay process and a periodic inventory system cost process. Results show that for processes with strong correlation over many lags, as is typical in queue-delay processes and inventory processes, the achieved coverage of the constructed interval is closer to nominal levels than for the unweighted-batch-means method. The coverage and average half-length is not as greatly affected by the number of batches, and nonnormality of the weighted batch means is not as severe.