Some properties of BLUE in a linear model and canonical correlations associated with linear transformations.

*(English)*Zbl 0731.62116Summary: 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 |

##### Keywords:

BLUE; covariance matrix; number of unit canonical correlations; g- inverse; nonzero canonical correlations; Moore-Penrose inverse
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\textit{C. G. Khatri}, J. Multivariate Anal. 34, No. 2, 211--226 (1990; Zbl 0731.62116)

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