ML estimation proceeds by minimizing the following fit function: FML(θ)=log∣Σ(θ)∣−log∣S∣+tr{SΣ−1(θ)}−k where Σ(θ) is the covariance matrix implied by the set of parameter estimates. Note that, while the formulation of the ML fit function does not explicitly include a weight matrix, it is asymptotically equivalent to a more general formulation that is identical to the WLS fit function, with .5Dk′(Σ−1(θ~)⊗Σ−1(θ~))Dk, where Dk is the duplication matrix of order k, in place of DS . Thus, the difference between ML and WLS estimation can be construed as a difference in weight matrices only. A comparison between ML and WLS results can be found in the Supplementary Results (see also Supplementary Figures 23–27, Supplementary Table 19).
