Correlation among estimators of the variance of the sample mean

Various types of estimators have been proposed for estimating the variance of the sample mean, a fundamental quantity in simulation output analysis. When used with low degrees of freedom, several of these estimators have little bias. But the low degrees of freedom correspond to high variance. One approach to creating estimators with smaller variance while maintaining the negligible bias is to use linear combinations of known estimators. Whether linear combinations provide improved estimators — and, if so, the choice of estimators to be included in the linear combination — depends upon the correlations among the various estimators. Linear combinations of estimators having high positive correlation would provide little improvement while combinations of independent estimators would provide substantial gain. We investigate the correlation among four well-known estimators as a function of the type of stochastic process generating the data, the sample size, the estimator type, and estimator parameters.