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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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