Fundamental Equations of BLUE and BLUP in the Multivariate Linear Model with Applications

Abstract

Ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE), and best linear unbiased predictor (BLUP) in the general linear model with new observations are generalized to the general multivariate linear model. The fundamental equations of BLUE and BLUP in the multivariate linear model are derived by two methods, including the vectorization method and projection method. By using the matrix rank method, some new results of linear BLUE-sufficiency, linear BLUP-sufficiency, and the equality of OLSE, BLUE, and BLUP are given in the multivariate linear model.

Keywords:

• BLUE
• BLUP
• Linear BLUE-sufficiency
• Linear BLUP-sufficiency
• Multivariate linear model
• Matrix rank method
• OLSE
• Projection method
• Vectorization method

Mathematics Subject Classification:

• 62J50
• 62H12
• 15A09
• 15A24

Acknowledgments

The authors are grateful to two anonymous referees for their helpful comments and constructive suggestions to an earlier version of the article. This work was supported in part by Natural Science Foundation of China (11271259), Shanghai Municipal Natural Science Foundation 10ZR1420600, and Foundation of Shanghai Municipal Education Commission (10YZ184, 11ZZ182).