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To construct confidence regions, the asymptotic variance requires estimation. For IID sampling, is estimated by the sample covariance matrix, as discussed in Section 2.3. For MCMC sampling, a rich literature of estimators of is available including spectral variance [14, 15], regeneration‐based [16, 17], and initial sequence estimators [5]18–20]. Considering the size of modern simulation output, we recommend the computationally efficient batch means estimators.

The multivariate batch means estimator considers nonoverlapping batches and constructs a sample covariance matrix from the sample mean vectors of each batch. More formally, let , where is the number of batches, and is the batch sizes. For , define . The batch means estimator of is

Univariate and multivariate batch means estimators have been studied in MCMC and operations research literature [21–26]. Although the batch means estimator has desirable asymptotic properties, it suffers from underestimation in finite samples, particularly for slowly mixing Markov chains. Specifically, let

Then, Vats and Flegal [27] show (ignoring smaller order terms)

When the autocorrelation in the Markov chain is large, or is small, there is significant underestimation in . To combat this issue, Vats and Flegal [27] propose lugsail batch means estimators formed by a linear combination of two batch means estimators with different batch sizes. For and , the lugsail batch means estimator is

(4)

It is then easy to see

When , the finite‐sample bias is positive. Vats and Flegal [27] recommend and , which induces a positive bias of offsetting the original bias in the opposite direction. For , this estimator corresponds to the flat‐top batch means estimator of Liu and Flegal [28]. Under polynomial ergodicity and additional conditions on the batch size , the lugsail batch means estimators are strongly consistent [26].