alt="z left-parenthesis alpha right-parenthesis"/> be a quantile of a standard normal distribution possibly chosen to correct for simultaneous inference. Recall that
, let
denote the
th component of
. Further, let
denote the
th diagonal of
. Then
The volume of this hyperrectangular confidence region is
(3)
As more samples are obtained, and converge to 0 so that the variability in the estimator disappears. Sequential stopping rules in Section 5 will utilize this feature to terminate simulation.
4 Estimating
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
