Rank and inertia formulas for covariance matrices of BLUPs in general linear mixed models

Simultaneous optimal predictions under two seemingly unrelated linear random-effects models

“…This idea for constructing general vectors as given in (1.14) and (1.15) was first given in [26] who showed a lemma on optimization of a matrix function in the Löwner partial ordering and established a unified theory of linear estimations/predictions of all unknown parameters in general linear models with fixed or mixed effects; see also [28,Lemma 4.7]. In addition, the work on separate and simultaneous estimations/predictions of unknown parameters in different models can be found in [1,[3][4][5][6][10][11][12][13][14]16,25,30,[38][39][40]; some new results concerning simultaneous linear estimations/predictions of all unknown parameters in LREMs with original and future observations were obtained in [33,34] by solving certain constrained quadratic matrix-valued function optimization problems in the Löwner partial ordering.…”

“…To account for general prediction/estimation problems of unknown parameters in a given linear regression model, it is common practice to first adopt a feasible procedure for obtaining exact expressions of predictors/estimators of the unknown parameters in the model. Tian recently developed an analytical method in [33,34] to solve certain types of constrained quadratic matrix-valued function optimization problem in the Löwner partial ordering, and used the method to examine some simultaneous linear estimations/predictions of all unknown parameters in LREMs with original and future observations; see also [7,11,15,17,18,31,32,[35][36][37] for a series of related approaches.…”