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

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, ? =  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 Wu , Q. G. ( 1988 ). Several results on admissibility of a linear estimate of stochastic regression coefficients and parameters . Acta Mathemaica Applicatae Sinica 11 ( 1 ): 95 – 106 . (in Chinese)  Google Scholar]), Baksalary and Markiewicz (1990 Baksalary , J. K. , Markiewicz , A. ( 1990 ). Admissible linear estimators of an arbitrary vector of parametric functions in the general Gauss–Markov model . J. Stat. Plann. Infer. 26 : 161 – 171 . Google Scholar]), and Groß and Markiewicz (1999 Groß , J. , Markiewicz , A. ( 1999 ). On admissibility of linear estimators with respect to the mean square error matrix criterion under the general mixed linear model . Statistics 33 : 57 – 71 .Taylor & Francis Online] , Google Scholar]). In addition, we derive the corresponding results on the admissibility problem under the generalized MSE criterion.