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Two-stage regression model. (English) Zbl 0662.62057
A mixed linear model is characterized by the relations \(E(Y| \beta)=X\beta\), \(Var(Y| \vartheta)=\sum^{p}_{i=1}\vartheta_ iV_ i\), where Y is an n-dimensional random vector, \(\beta\) is an unknown k-dimensional parameter, \(\beta\in {\mathcal R}^ k\) (k-dimensional Euclidean space), X is a known \(n\times k\) matrix, \(\vartheta\) is a p- dimensional vector of variance components (usually unknown), \(\vartheta =(\vartheta_ 1,...,\vartheta_ p)'\in {\underline \Theta}\subset {\mathcal R}^ p\), \({\underline \Theta}\) is an open set, \(V_ i\), \(i=1,...,p\), are known symmetric \(n\times n\) matrices; E and Var denote mean value and covariance matrix, respectively. If \[ Y=(Y'_ 1,Y'_ 2)',\quad X=\left[ \begin{matrix} X_ 1\\ C\end{matrix} \begin{matrix} 0\\ X_ 2\end{matrix} \right],\quad C\neq 0,\quad \Sigma =\left[ \begin{matrix} \Sigma_{11}\\ 0\end{matrix} \begin{matrix} 0\\ \Sigma_{22}\end{matrix} \right], \] where the dimensions of the vectors \(Y_ 1\) and \(Y_ 2\) are \(n_ 1\) and \(n_ 2\) \((n_ 1+n_ 2=n)\), the matrices \(X_ 1\), \(X_ 2\) are of the types \(n_ 1\times k_ 1\), \(n_ 2\times k_ 2\) and the matrices \(\Sigma_{11}\), \(\Sigma_{22}\) are of the types \(n_ 1\times n_ 1\), \(n_ 2\times n_ 2\), respectively, then the regression model (Y,X\(\beta\),\(\Sigma)\) is called the two-stage regression model.
The aim of the paper is to find the locally (or uniformly) best estimators of the parameters \(\beta\) and \(\vartheta\) under some specified conditions.

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