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A lemma on optimization of a matrix function and a review of the unified theory of linear estimation. (English) Zbl 0735.62066
Statistical data analysis and inference, Pap. Int. Conf., Neuchâtel/Switzerl. 1989, 397-417 (1989).
Summary: [For the entire collection see Zbl 0732.00019.]
A method of computing the minimum of the matrix valued function $$A'VA- F'A-A'F$$ of a matrix argument $$A$$ subject to a constraint of the form $$X'A=P$$ is proposed. The solution is used to develop a unified theory of estimation in the mixed effects linear model $$\underset{\widetilde{\phantom{m}}} Y=X\underset{\widetilde{\phantom{m}}}\beta +U\underset{\widetilde{\phantom{m}}}\xi+\epsilon$$ without making any assumptions on the ranks of the matrices $$X$$, $$U$$ or of the dispersion matrices of $$\underset{\widetilde{\phantom{m}}}\xi$$ and $$\underset{\widetilde{\phantom{m}}}\epsilon$$. The estimates of $$X\underset{\widetilde{\phantom{m}}}\beta$$, $$U\underset{\widetilde{\phantom{m}}}\xi$$ and $$\underset{\widetilde{\phantom{m}}}\epsilon$$ depend on the elements of a $$g$$-inverse of a partitioned matrix as in the case of the fixed effects model.
A new concept of conditioned equations (similar to that of normal equations) is introduced for simultaneous estimation of fixed effects $$(X\underset{\widetilde{\phantom{m}}}\beta)$$, random effects $$(U\underset{\widetilde{\phantom{m}}}\xi)$$, and the error $$(\underset{\widetilde{\phantom{m}}}\epsilon)$$. The paper is self- contained and simplifies the approach presented in earlier papers of the author on a unified theory of linear estimation based on minimizing a quadratic form.

##### MSC:
 62J05 Linear regression; mixed models 15A24 Matrix equations and identities 15A09 Theory of matrix inversion and generalized inverses