UC Riverside

Variance estimation in the context of high dimensional Markov Chain Monte Carlo (MCMC) is an interesting topic in research. Practical implications of estimating variance are limited due to inherent systematic bias both in univariate and multivariate settings. Recent advancements in high dimensional covariance matrix estimation in the MCMC setting, including works on Lugsail Batch Means (LUG-BM), have proven to improve the bias properties. Using spectral theory of estimation, we can further improve upon the bias and variance properties of the estimators. Finite sample properties of variance estimators should be studied in detail using statistical properties of the sampling bias, while accounting for the sampling error. The direction of this bias is crucial in finite sample applications. Mean Square Error (MSE) has been traditionally used to assess the quality of estimation. However, using alternate asymmetrical loss functions is recommended as they are more natural to use in applications where constructing optimal variance estimators in finite samples is required. Normality assumptions are required to make efficient use of these techniques and a careful analysis should be done to ensure the assumptions for normality are met.