Variance-covariance component estimation based on the equivalent residuals

The development of the variance-covariance component estimation(VCE)theory is firstly synoptically reviewed in this paper.Then the equivalent residuals are extracted by using orthogonal decomposition and the fundamental equations for VCE are established.Based on that the two profound and unresolvable problems for VCE theory,namely regional optimality and negative definition for estimated covariance matrix,are explored and the corresponding possible resolvable schemes and their complexity are analysed.Thirdly,we derive out the Helmert,least squares and MINQUE VCE formulae based on the fundamental equations with the given initial values,and additionally we also prove their equivalence with the existing VCE formulae.The procedure of derivation is beneficial for us to understand the essence of VCE that all VCE formulae are identical.Finally,two examples are performed to verify the proposed viewpoints.