Читать книгу Computational Statistics in Data Science - Группа авторов - Страница 127

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

(2)

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 . 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.