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Additional resources for Analysis of variance for random models, vol.2: Unbalanced data
Finally, it should be noted that Henderson’s methods may produce negative estimates. Khattree (1998, 1999) proposed some simple modiﬁcations of Henderson’s procedures which ensure the nonnegativity of the estimates. The modiﬁcations entail seeking nonnegative estimates to Henderson’s solution that are closest to the expected values of the quadratics being used for estimation. The resulting estimators are found to be superior in terms of various comparison criteria to Henderson’s estimators except in the case of the error variance component.
Vandaele and Chowdhury (1971) proposed a revised method of scoring that will ensure convergence to a local maximum of the likelihood function, but there is no guarantee that the global maximum will be attained. Hemmerle and Hartley (1973) discussed the Newton–Raphson method for the mixed model estimation which is closely related to the method of scoring. Jennrich and Sampson (1976) presented a uniﬁed approach of the Newton–Raphson and scoring algorithms to the estimation and testing in the general mixed model analysis of variance and discussed their advantages and disadvantages.
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. 239–244), Searle et al. (1992, pp. 234–237), and Rao (1997, pp. 93–96). 1). Necessary and sufﬁcient conditions for the existence of ML estimates of the variance components are considered by Demidenko and Massaam (1999).