The Song rule outperforms optimal-batch-size variance estimators in simulation output analysis

Abstract

Estimating the variance of the sample mean is a fundamental problem in Monte Carlo simulation output analysis. The need is to develop a procedure to estimate this variance with minimum mean-squared-error (mse). One of the commonly used approaches is batch-means estimator (BME) including non-overlapping batch means (NBM) and overlapping batch means (OBM). The research into BMEs has pursued the elusive optimal-batch-size for many years. Another commonly used approach is to linearly combine two BMEs with large batch sizes to ignore estimating the bias constant. This paper demonstrates that such two types of pursuits are not the optimal surrogate for the minimum mse.This research proposes an analytical variance estimator which linearly combines OBM with batch size 2 and NBM with batch size 1, and having the optimal weight written as a function of the sum of all correlations of the process (γ0). The proposed rule, called the Song rule, captures 4 keywords: Smallest-batch-sizes; OBM(m=2); NBM(m=1); and Gamma0 (γ0). The proposed rule has a solid theoretical base and dramatically outperforms the optimal-batch-size OBM. This paper also provides an implementable version, which reduces mse above 45% for all cases studied, comparing with OBM with the associated optimal-batch-size, rather than its estimated optimal-batch-size. Both of the proposed analytical and implementable rules provide significant advancements in the estimation of the variance of the sample mean.