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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.

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