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Linear estimation with an incorrect dispersion matrix in linear models with a common linear part. (English) Zbl 0536.62052
The linear model (Y,\(X\beta\),V) where \(Y=X\beta +e\) and \(V=(cov(e))\) is considered. For any matrix A, Let \({\mathcal R}(A)\) denote the vector space spanned by the columns (or the range space); r(A) denotes the rank of A. The article starts with the set up considering all linear models \((Y,X\beta,V_ 1)\) with a common linear part \({\mathcal R}(U)\), that is \(r(X)=p<n\) such that \({\mathcal R}(U)\subset {\mathcal R}(X)\), U being a known matrix with \(r(U)<p\). The class of such matrices is denoted by \(C^ p(U)\). The considered design matrices \(X\in C^ p(U)\) further satisfy the condition \({\mathcal R}(A^ t)\subset {\mathcal R}(X^ t)\), where A is a given matrix with 1\(\leq r(A)\leq p\). The class of such matrices is denoted by \(C_ A^ p(U).\)
The author characterizes matrices V such that every linear representation or some linear representation of the BLUE of \(A\beta\) under \((Y,X\beta,V_ 1)\) continues to be its BLUE under (Y,\(X\beta\),V) also. Further on, variance components and covariance models are dealt with. Three conditions are proved under which \(A\beta\) admits a BLUE for every \(X\in C_ A^ p(U)\).
Reviewer: B.Kolleck

MSC:
62J05 Linear regression; mixed models
62J10 Analysis of variance and covariance (ANOVA)
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