Optimal mean-squared-error batch sizes

When an estimator of the variance of the sample mean is parameterized by batch size, one approach for selecting batch size is to pursue the minimal mean squared error (mse). We show that the convergence rate of the variance of the sample mean, and the bias of estimators of the variance of the sample mean, asymptotically depend on the data process only through its marginal variance and the sum of the autocorrelations weighted by their absolute lags. Combining these results with variance results of Goldsman and Meketon, we obtain explicit asymptotic approximations for mse, optimal batch size, optimal mse, and robustness for four quadratic-form estimators of the variance of the sample mean. Our empirical results indicate that the asymptotic approximations are reasonably accurate for sample sizes seen in practice. Although we do not discuss batch-size estimation procedures, the empirical results suggest that the explicit asymptotic batch-size approximation, which depends only on a summary measure (which we refer to as the balance point) of the nonnegative-lag autocorrelations, is a reasonable foundation for such procedures.

[1]  D. Daley The serial correlation coefficients of waiting times in a stationary single server queue , 1968, Journal of the Australian Mathematical Society.

[2]  George S. Fishman,et al.  Estimating Sample Size in Computing Simulation Experiments , 1971 .

[3]  D. Brillinger Estimation of the mean of a stationary time series by sampling , 1973, Journal of Applied Probability.

[4]  Michael A. Crane,et al.  Simulating Stable Stochastic Systems: III. Regenerative Processes and Discrete-Event Simulations , 1975, Oper. Res..

[5]  VARIANCE REDUCTION TECHNIQUES FOR THE SIMULATION OF QUEUEING NETWORKS , 1979 .

[6]  Philip Heidelberger,et al.  A spectral method for confidence interval generation and run length control in simulations , 1981, CACM.

[7]  Bruce W. Schmeiser,et al.  Batch Size Effects in the Analysis of Simulation Output , 1982, Oper. Res..

[8]  Lee W. Schruben,et al.  Confidence Interval Estimation Using Standardized Time Series , 1983, Oper. Res..

[9]  D. B. Preston Spectral Analysis and Time Series , 1983 .

[10]  R. W. Andrews,et al.  Arma-Based Confidence Intervals for Simulation Output Analysis , 1984 .

[11]  Bruce W. Schmeiser,et al.  Overlapping batch means: something for nothing? , 1984, WSC '84.

[12]  Peter D. Welch,et al.  On the relationship between batch means, overlapping means and spectral estimation , 1987, WSC '87.

[13]  Bruce W. Schmeiser,et al.  Correlation among estimators of the variance of the sample mean , 1987, WSC '87.

[14]  Bruce W. Schmeiser,et al.  On the dispersion matrix of variance estimators of the sample mean in the analysis of simulation output , 1988 .

[15]  B. Schmiser,et al.  Inverse-transformation algorithms for some common stochastic processes , 1989, WSC '89.

[16]  U. Narayan Bhat,et al.  A Sequential Inspection Plan for Markov Dependent Production Processes , 1990 .

[17]  Keebom Kang,et al.  An Investigation of Finite-Sample Behavior of Confidence Interval Estimators , 1992, Oper. Res..

[18]  Bruce W. Schmeiser,et al.  Variance of the Sample Mean: Properties and Graphs of Quadratic-Form Estimators , 1993, Oper. Res..

[19]  Halim Damerdji,et al.  Mean-Square Consistency of the Variance Estimator in Steady-State Simulation Output Analysis , 1995, Oper. Res..

[20]  Wheyming Tina Song,et al.  On the estimation of optimal batch sizes in the analysis of simulation output , 1996 .