Let {(yit, Eit) ∈ ℠x ℠| t = 1, ..., k} be a sample of observations satisfying the random intercepts model Yit = Qi + Brit + Eit, where α1 and α2 are iid. N(α, σ0^2) random variables; {Eit | i = 1, ..., k, t = 1, ..., k} are iid. N(0, σ^2) random variables; and α1, α2, and Eit's are independent of each other. We assume Cit's are fixed; (α, β) are unknown parameters; and σ0^2 and σ^2 are variances of known values.
Find E(yit) and Var(yit). Show that Cov(yit, Yis) = 0 when t ≠s. Show that random vectors Y1 = (y11, ..., y1k)T and Y2 = (y21, ..., y2k)T are independent, and Yi follows a multivariate normal distribution MVN((1, X)(α, β)T, σ^2Ik + σ0^2Ek), where 1 is a k x 1 vector of 1's, X = (T, T)Ik is a k x k identity matrix, and Ek is a k x k matrix of 1's. Write down the joint pdf of y1 and Y2. Then write down the log-likelihood function of (α, β) for the observed data. Finally, show that the MLE of (α, β) is (α, β)T = (C(X)T(C(X)C(X)T)^-1C(X)T)^-1C(X)T(Y), which is a weighted least-squares solution. (Hint: Note that (Ik + σ0^2Ek)^-1 = Ik - σ0^2Ek/(σ^2 + σ0^2k).)