Computational Statistics in Data Science. Группа авторов

      We obtain the asymptotic distribution of . A similar argument can be made for the off‐diagonals of . Under the conditions of Theorem 1,

      where is

      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

(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