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Three rank formulas associated with the covariance matrices of the BLUE and the OLSE in the general linear model. (English) Zbl 1072.62049
Summary: We consider the estimation of the expectation vector $$X\beta$$ under the general linear model $$\{ y,X\beta,\sigma^2V\}$$. We introduce a new handy representation for the rank of the difference of the covariance matrices of the ordinary least squares estimator OLSE($$X\beta$$) (= $$Hy$$, say) and the best linear unbiased estimator BLUE($$X\beta$$) (= $$Gy$$, say). From this formula, some well-known conditions for the equality between $$Hy$$ and $$Gy$$ follow at once. We recall that the equality between $$Hy$$ and $$Gy$$ can be characterized by the rank-subtractivity ordering between the covariance matrices of $$y$$ and $$Hy$$. This rank characterization suggests a particular presentation for the rank of the difference of the covariance matrices of $$Hy$$ and $$Gy$$. We show, however, that this presentation is valid if and only if the model is connected.

##### MSC:
 62H12 Estimation in multivariate analysis 62J05 Linear regression; mixed models
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##### References:
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