MINQUE of Variance-Covariance Components in Linear Gauss-Markov Models

For heterogeneous and correlated observations, the variance components and the covariance components sometimes must be estimated. The forms of best invariant quadratic unbiased estimate (BIQUE) and Helmert-type estimation of variance and covariance components have already been derived by Koch and Grafarend, respectively. After obtaining the minimum norm quadratic unbiased estimate (MINQUE) of variance components, Rao derived only the MINQUE of the variance and covariance components for a special case in which the error vector is composed of a linear combination of independent random effect vectors of zero mean and the same variance-covariance matrix whose variance and covariance components were to be determined. However, an explicit expression of the MINQUE suitable to more general situations has not been derived. This paper defines the natural estimation of covariance components from errors and derives the MINQUE of variance and covariance components. The BIQUE and MINQUE of variance components without covariance components have the same iteration solution; the Helmert solution is only a special case of the MINQUE. However, the three estimates of variance and covariance components are different. The two MINQUE methods obtained in this paper have the advantage independence of the error distribution and offer a reasonable alternative in estimating variance and covariance components, and they can be used in the most general case. Numeric results show that the two MINQUE methods obtained in this paper are feasible.