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Some properties of BLUE in a linear model and canonical correlations associated with linear transformations. (English) Zbl 0731.62116
Summary: Let (x,X$$\beta$$,V) be a linear model and let $$A'=(A'_ 1,A'_ 2)$$ be a $$p\times p$$ nonsingular matrix such that $$A_ 2X=0$$, Rank $$A_ 2=p-Rank X$$. We represent the BLUE and its covariance matrix in alternative forms under the conditions that the number of unit canonical correlations between $$y_ 1$$ $$(=A_ 1x)$$ and $$y_ 2$$ $$(=A_ 2x)$$ is zero. For the second problem, let $$x'=(x'_ 1,x'_ 2)$$ and let a g-inverse $$V^-$$ of V be written as $$(V^-)'=(A'_ 1,A'_ 2)$$. We investigate the reactions (if any) between the nonzero canonical correlations $$\{1\geq \rho_ 1\geq...\geq \rho_ i>0\}$$ due to $$y_ 1$$ $$(=A_ 1x)$$ and $$y_ 2$$ $$(=A_ 2x)$$, and the nonzero canonical correlations $$\{1\geq \lambda_ 1\geq...\geq \lambda_{v+r}>0\}$$ due to $$x_ 1$$ and $$x_ 2$$. We answer some of the questions raised by D. Latour et al. at the 2nd Int. Tampere Conf. Statist. (1987) in the case of the Moore-Penrose inverse $$V^+=(A'_ 1,A'_ 2)$$ of V.

##### MSC:
 62H20 Measures of association (correlation, canonical correlation, etc.) 62J05 Linear regression; mixed models 15A09 Theory of matrix inversion and generalized inverses
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