Admissible Estimation for Linear Combination of Fixed and Random Effects in General Mixed Linear Models

Abstract

The general mixed linear model can be denoted by y  =  X β +  Z u  +  e , where β is a vector of fixed effects, u is a vector of random effects, and e is a vector of random errors. In this article, the problem of admissibility of Q y and Q y  +  q for estimating linear functions, ϱ =  L ′β +  M ′ u , of the fixed and random effects is considered, and the necessary and sufficient conditions for Q y (resp. Q y  +  q ) to be admissible in the set of homogeneous (resp. potentially inhomogeneous) linear estimators with respect to the MSE and MSEM criteria are investigated. We provide a straightforward alternative proof to the method that was utilized by Wu (1988), Baksalary and Markiewicz (1990), and Groß and Markiewicz (1999). In addition, we derive the corresponding results on the admissibility problem under the generalized MSE criterion.

Keywords:

• Admissibility
• General mixed linear model
• (Generalized) MSE criterion
• Linear combination
• MSEM criterion

Mathematics Subject Classification:

• 62C15
• 62J05

Acknowledgments

This work is mainly motivated by the works of Wu (1983, 1986, 1987, 1988), Yu and Xu (2004), and Xu and Yu (2005). The author is very grateful to them for stimulating aspects. This research was supported by Grants HGC0923 and HGC0925 and the “Green & Blue Project” Program for 2008 to Cultivate Young Core Instructors from Huaiyin Institute of Technology.