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Estimation of variance and covariance components - MINQUE theory. (English) Zbl 0223.62086
J. Multivariate Anal. 1, 257-275 (1971).
The paper consists of two parts. The first part deals with solutions to some optimization problems. The general problem is one of minimising trace $$AVA' U$$ with respect to elements of matrix $$A$$, where $$V$$ and $$U$$ are positive definite matrices, subject restrictions of the type $$AX=0$$ or $$X'AX=0$$ and trace $$AV_i=p_i$$, $$i=1,\dots,k$$, or $$U_1'AU_1+\dots+U_k'AU_k=M$$ where $$V_i, U_i, M, p_i$$ are given. Two situations are considered, when $$A$$ is a general $$m\times n$$ matrix and when $$A$$ is restricted to the class of symmetric $$n\times n$$ matrices.
The results are applied in the proposed theory of estimation of variance components called MINQUE (minimum norm quadratic unbiased estimation). We consider the linear model $$Y=X\beta+\varepsilon$$ where $$E(\varepsilon)=0$$ and $$D(\varepsilon)=\sigma_1^2V_1+\dots+\sigma_k^2V_k$$, where $$V_i$$ are known and $$\sigma_i^2$$ are to be estimated. The quadratic statistic $$Y'AY$$ is said to be MINQUE of the parametric function $$p_1\sigma_1^2+\dots+p_k\sigma_k^2$$ if $$A$$ such that $$AX=0$$ or $$X'AX=0$$ and trace $$AV_i=p_i$$, $$i=1,\dots,k$$, and subject to these conditions $$AVAv$$ is a minimum. Two choices of $$V$$ are suggested: $$V=V_1+\dots+V_k$$ and $$V=\alpha_1V_1+\dots+\alpha_kV_k$$ where $$\alpha_1,\dots,\alpha_k$$ are apriori values of $$\sigma_1^2,\dots,\sigma_k^2$$. The paper also considers the estimation of the covariance matrix $$\Sigma$$ when $$D(\varepsilon) = U_1'\Sigma U_1 + \dots + U_k'\Sigma U_k$$in the linear model.