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V k ). Suppose also that V i = V i (µi , λi ), i = 1, 2, . . , k where µi = (µi1 , . . , µini ) is the vector of means for the ith individual and λi is a parameter including variance and correlation components. , µit = µit (θ), t = 1, 2, . . , ni , i = 1, 2, . . , k. For example, in the case of binary response variables and ni = T for each i, we may suppose that P Yit = 1 xit , θ = µit , log µit (1 − µit ) = xit θ, where xit represents a covariate vector associated with individual i and time t.

2 We shall herein drop the subscript T for convenience. (i) The condition that E(G∗ ) − E(G) is nnd immediately gives tr (E(G∗ ) − E(G)) = tr E(G∗ ) − tr E(G) ≥ 0. Conversely, suppose H satisﬁes tr E(H) ≥ tr E(G) for all G ∈ H. If there is an OF -optimal G∗ , then tr E(H) ≥ tr E(G∗ ). But from the deﬁnition of OF optimality we also have tr E(G∗ ) ≥ tr E(H) and hence tr E(G∗ ) = tr E(H). Thus, we have that A = E(G∗ ) − E(H) is nnd and tr A = 0. But A being symmetric and nnd implies that all its eigenvalues are positive, while tr A = 0 implies that the sum of all the eigenvalues of A is zero.

An application of the Simultaneous Reduction Lemma to the pair E(G∗ ), E(H), the former taken as pd, leads immediately to E(G∗ ) = E(H) and an OF -optimal solution. Remark It must be emphasized that the existence of an OF -optimal estimating function within H is a crucial assumption in the theorem. For example, if G∗ satisﬁes the trace criterion (i), it is not ensured that G∗ is an OF -optimal estimating function within H; there may not be one. 1 The Framework Historically there have been two distinct approaches to parameter inference developed from both classical least squares and maximum likelihood methods.