Bias-Aware Linear Combinations of Variance Estimators

摘要

A prototype problem in the analysis of steady-state stochastic processes is that of estimating the variance of the sample mean. A commonly used performance criterion for variance estimators is the mean-squared-error (mse) - the sum of the variance and the squared bias. In this paper, we attempt to minimize the variance of an estimator subject to a bias constraint - a goal that differs from that of minimizing mse, in which case there would be no explicit bias constraint. We propose a bias-aware mechanism to achieve our goal. Specifically, we use linear combinations of estimators based on different batch sizes to approximately satisfy the bias constraint; and then we minimize the variance by choosing appropriate linear combination weights. We illustrate the use of this mechanism by presenting bias-aware linear combinations of several variance estimators, including non-overlapping batch means, overlapping batch means, and standardized time series weighted area estimators. We also evaluate our mechanism with Monte Carlo examples.