Part IV’s primary question is this: What functions of the data supply information about each of the unknowns in the mixed linear model’s covariance matrices G and R?
For example, in the balanced one-way random effects model with yi j = µ+ui+ εi j, i = 1, . . . ,N, j = 1, . . . ,m, where ui ∼ N(0,σ2s ) and εi j ∼ N(0,σ2e ), how do the data provide distinct information about σ2s and σ2e ? For σ2e , yi j− y¯i. ∼ N(0, m−1m σ2e ), where y¯i. is the ith group’s average, so these functions of the data provide “clean” information about σ2e . What about σ2s ? It appears that no function of the data provides information about σ2s that is not “contaminated” by σ2e . But the plot thickens: Because var(y¯i.) = σ2s +σ2e /m, the group averages appear to provide some information about σ2e . So even this simplest case is not so simple.1