Second-Order Efficiency for Two-Stage Estimation of a Linear Function of Normal Mean Vectors when Covariance Matrices Have Some Structures

Abstract

We consider two-stage estimation for a fixed-span confidence region about a linear function of mean vectors from π _i : N _p (μ _i , Σ _i ), i = 1,…,k (≥2), when Σ _i 's have some structures. The purpose of this article is to investigate asymptotic efficiency of the estimation up to the second order in terms of the sample size. An adjustment of the design constant and a proper choice of the initial sample size appearing in the two-stage estimation are proposed to have asymptotic second-order efficiency. Some simulations are carried out to see moderate sample size performances of the proposed two-stage estimation. An example is given for a demonstration.

Keywords:

• Asymptotic consistency
• Covariance structure
• Fixed-span confidence region
• Sample size
• Second-order efficiency
• Two-stage procedure

Subject Classification:

• 62L10
• 62L12

ACKNOWLEDGMENTS

We would like to thank the anonymous referee and the Editor, Professor Nitis Mukhopadhyay, for their valuable comments. Research of the first author was partially supported by a Grant-in-Aid for Exploratory Research, the Ministry of Education, Culture, Sports, Science, and Technology, Japan, under contract number 16650059.

Notes

Recommended by Wei Liu