Properties of batched quadratic-form variance parameter estimators for simulations

We examine the practice of batching of certain quadratic-form estimators for the variance parameter of a stochastic process. The class of batched quadratic-form estimators includes, among others, the standardized time series (STS) weighted area and weighted Cramér-von Mises estimators. We give results on the expected value and variance of such estimators as the batch size and/or the number of batches increase. In particular, we show that the above STS estimators are consistent for the variance parameter in terms of mean squared error. An analytical example involving a first-order autoregressive process illustrates our findings.