The correlation between mean and variance estimators

One of the pitfalls encountered when using confidence interval estimators for the mean of a stationary stochastic process is that the mean and variance estimators (X and V) may be correlated. We derive Corr (X, V) for various variance estimators and stochastic processes, and we examine the effects of this correlation upon confidence interval estimator performance. Among the variance estimators under consideration are those arising from the methods of batched means and standardized time series. Both small sample and asymptotic results are reported.