A new algebraic analysis to linear mixed models
نویسنده
چکیده
This article presents a new investigation to the linear mixed model y = Xβ + Zγ + ε with fixed effect Xβ and random effect Zγ under a general assumption via some novel algebraic tools in matrix theory, and reveals a variety of deep and profound properties hidden behind the linear mixed model. We first derive exact formulas for calculating the best linear unbiased predictor (BLUP) of a general vector φ = Fβ + Gγ + Hε of all unknown parameters in the model by solving a constrained quadratic matrix-valued function optimization problem in the Löwner partial ordering. We then consider some special cases of the BLUP for different choices of F, G, and H in φ, and establish some fundamental decomposition equalities for the observed random vector y and its covariance matrix. Mathematics Subject Classifications: 62H12; 62J05
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