Second-Order Properties of a Two-Stage Fixed-Size Confidence Region for the Mean Vector of a Multivariate Normal Distribution

We consider the classical fixed-size confidence region estimation problem for the mean vectorμin theNp(μ, Σ) population where Σ is unknown but positive definite. We writeλ1for the largest characteristic root of Σ and assume thatλ1is simple. Moreover, we suppose that, in many practical applications, we will often have available a numberλ*(>0) and that we can assumeλ1>λ*. Given this addi- tional, and yet very minimal, knowledge regardingλ1, the two-stage procedure of Chatterjee (Calcutta Statist. Assoc. Bull.8(1959a), 121–148;9(1959b), 20–28;11(1962), 144–159) is revised appropriately. The highlight in this paper involves the verification ofsecond-order propertiesassociated with such revised two-stage estimation techniques, along with the maintenance of the nominal confidence coefficient.