Estimation of variance and covariance components - MINQUE theory. (English) Zbl 0223.62086

Rao, C. Radhakrishna

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.

Reviewer: C.Radhakrishna Rao

MSC:

62J10

Analysis of variance and covariance (ANOVA)

Keywords:

Estimation; variance components; covariance components; MINQUE theory; Minimum Norm Quadratic Unbiased Estimators (MINQUE)

References:

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