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.


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