Double shrink methodologies to determine the sample size via covariance structures

We consider fixed-size estimation for a linear function of mean vectors from π i : N p ( μ i , Σ i ) , i = 1 , … , k , when every Σ i has some structure. The goal of inference is to construct a fixed-span confidence region with required accuracy. We find a sample size for each π i with the help of the ‘double shrink methodology’, that is introduced by this paper, via covariance structures of Σ i , i = 1 , … , k . We estimate the sample size in a two-stage sampling and give a fixed-span confidence region that has the coverage probability approximately second-order consistent with the required accuracy. Some simulations are carried out to see moderate sample size performances of the proposed methodologies.