By H. Sahai, M. Ojeda
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Extra resources for Analysis of Variance for Random Models [Vol II - Unbalanced Data]
6) have to be solved for the elements of α, σe2 , and the ρi s contained in H with the constraints that the σe2 and ρi s be nonnegative. 6), which are difﬁcult equations to handle. The difﬁculty arises because the ML equations may yield multiple roots or the ML estimates may be on the boundary points. 5) can be readily solved in terms of ρi s. 7a) and 1 ˆ H −1 (Y − Xα) ˆ (Y − Xα) N 1 = [Y H −1 Y − (X H −1 Y ) (X H −1 X)−1 (X H −1 Y )]. 8) where R = I − X(X H −1 X)−1 X H −1 . 8). For some alternative formulations of the likelihood functions and the ML equations, see Hocking (1985, pp.
21), the weights σi2 s are, of course, unknown. Rao (1972) suggested the following two amendments to this problem: (i) If we have a priori knowledge of the approximate ratios σ12 /σp2 , . . 21) and use the W thus comσp−1 p puted. 21) and obtain MINQUEs of σi2 s. 21) and the MINQUE procedure repeated. 1). 10. Minimum-Norm/-Variance Quadratic Unbiased Estimation 45 the property of unbiasedness is usually lost; but the estimates thus obtained may have some other interesting properties. Rao (1971a) also gives the conditions under which the MINQUE is independent of a priori weights σi2 s.
2 Y Qp Y = σp tr(Qp Vp ). Note that the procedure depends on the order of the Uj s in the deﬁnition of the projection operators Pi s. (ii) For completely nested random models, Henderson’s Methods I, II, and III reduce to the customary analysis of variance procedure. 22 Chapter 10. Making Inferences about Variance Components (iii) A general procedure for the calculation of expected mean squares for the analysis of variance based on least squares ﬁtting constants quadratics using the Abbreviated Doolittle and Square Root methods has been given by Gaylor et al.
Analysis of Variance for Random Models [Vol II - Unbalanced Data] by H. Sahai, M. Ojeda