Cramér-von Mises Variance Estimators for Simulations

We study estimators for the variance parameter σ² of a stationary process. The estimators are based on weighted Cramér-von Mises statistics, and certain weightings yield estimators that are “first-order unbiased” for σ². We derive an expression for the asymptotic variance of the new estimators; this expression is then used to obtain the first-order unbiased estimator having the smallest variance among fixed-degree polynomial weighting functions. Our work is based on asymptotic theory; however, we present exact and empirical examples to demonstrate the new estimators' small-sample robustness. We use a single batch of observations to derive the estimators' asymptotic properties, and then we compare the new estimators among one another. In real-life applications, one would use more than one batch; we indicate how this generalization can be carried out.