In this paper it is shown that the Heteroscedastic Method II based on Healy’s two-stage sampling is more economical than the original HM with Chatterjee’s two-stage sampling in simultaneous inference for k mean vectors of Np(μj,Σj), j=1,...,k, Σj’s being completely unknown (hence possibly unequal). The paper provides by using each procedure, sets of simultaneous confidence intervals on trA'M, A∈ℝpk such that trA'A=1, M=[μ1,...,μk] for: (i) any A, (ii) A 1=0, and (iii) A'1=0, where 1=(1,...,1)'. Each set meets requirements of prespecified common length and a prespecified simultaneous confidence coefficient. It derives an asymptotic expansion formula for the upper percentile point of the statistic in the formulation for each set; this determines the total sample size for each set to satisfy the requirements. It is verified that HM-II has smaller sample sizes than does HM, and yet meets the same requirements in the heteroscedastic simultaneous inference even if the initial sample size is finite.
