Asymptotic Second-Order Efficiency for Multivariate Two-Stage Estimation of a Linear Function of Normal Mean Vectors

We consider an asymptotic second-order efficiency of two-stage estimation for a fixed-span confidence region about a linear function of normal mean vectors from . It is shown that when p ≥ 3, no matter how the initial sample size is chosen, the two-stage estimation does not become asymptotically second-order efficient even under the assumption that a known lower bound is available for the maximum latent root of unknown . An adjustment of the design constant and a proper choice of the initial sample size, appeared in the two-stage estimation, are proposed to enjoy possessing the asymptotic second-order efficiency under that assumption as well as the asymptotic consistency. Numerical examples show that the proposed method reduces sample sizes significantly in the estimation with a guaranteed high accuracy.