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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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