## Quadratic estimations in mixed linear models.(English)Zbl 0742.62074

Given the linear model $$E\{Y\}=X_ 1\beta_ 1+\dots+X_ p\beta_ p=X\beta$$ and $$D\{Y\}=V_ 1\theta_ 1+\dots+V_ q\theta_ q$$ of first order fixed parameter unknowns $$[\beta_ 1,\dots,\beta_ p]=\beta'$$ and second order fixed parameter unknowns (variance components) $$[\theta_ 1,\dots,\theta_ q]=\theta'$$ of an $$n\times 1$$ random vector $$Y'$$ of observations, the paper deals with minimum norm quadratic (unbiased) estimations of linear functions of variance components $$q=f_ 1\theta_ 1+\dots+f_ q\theta_ q$$. The following four estimations $$\hat q$$ of the type $\hat q(Y,V,S)=Y'A(V,S)Y,$ (i) MINQE(S), (ii) invariant for translations in $$\beta$$: MINQE(I,S), (iii) unbiased: MINQE(U,S), (iv) MINQE(U,I,S), are described by 5 theorems and 4 corollaries. An illustrative example for the estimations (i)-(iv) of two independent observations $$y_ 1$$, $$y_ 2$$ of unknown mean value $$E\{Y\}$$ and different variances $$v(y_ 1)=\theta_ 1$$ and $$v(y_ 2)=\theta_ 2$$ is given.

### MSC:

 62J10 Analysis of variance and covariance (ANOVA) 62J99 Linear inference, regression
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### References:

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