Under IID sampling, the infinite sum above reduces to
Applying the delta method for function , we obtain
3.4 Confidence Regions for Means
Suppose that is an estimate of the limiting Monte Carlo variance–covariance matrix, for IID sampling, and for MCMC sampling. Let be the ‐quantile of a distribution. The CLT yields a large‐sample confidence region around as
Let denote the determinant. The volume of this ellipsoidal confidence region, which depends on , , and , is given by
Sometimes a joint sampling distribution may be difficult to obtain, or the limiting variance–covariance matrix may be too complicated to estimate. In such cases, one can consider hyperrectangular confidence regions. Let 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
1,
