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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.

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