Double shrink methodologies to determine the sample size via covariance structures

We consider fixed-size estimation for a linear function of mean vectors from πi:Np(μi,Σi)πi:Np(μi,Σi), i=1,…,ki=1,…,k, when every ΣiΣi 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 πiπi with the help of the ‘double shrink methodology’, that is introduced by this paper, via covariance structures of ΣiΣi, i=1,…,ki=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.