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.

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:

[1] | Focke, J.; Dewess, G., Über die schätzmethode MINQUE von C. R., (1971), Rao und ihre Verallgemeinerung, in press · Zbl 0289.62024 |

[2] | Hartley, H.O.; Rao, J.N.K., Maximum likelihood estimation for the mixed analysis of variance model, Biometrika, 54, 99-108, (1967) · Zbl 0178.22001 |

[3] | Mitra, S.K., Another look at Rao’s MINQUE of variance and covariance components, () · Zbl 0263.62042 |

[4] | Rao, C.Radhakrishna, () |

[5] | Rao, C.Radhakrishna, Calculus of generalized inverse of matrices, part 1: general theory, Sankhyā ser. A, 29, 317-342, (1967) · Zbl 0178.03103 |

[6] | Rao, C.Radhakrishna, Estimation of variance and covariance components in linear models, (), (in press, J.A.S.A.) · Zbl 0231.62082 |

[7] | Rao, C.Radhakrishna, Estimation of heteroscedastic variances in linear models, J.a.s.a., 65, 161-172, (1970) |

[8] | Rao, C.R.; Mitra, S.K., (), in press |

[9] | Searle, S.R., Topics in variance component estimation, Biometrics, 27, 1-76, (1971) |

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