4.15 For the one-way layout with random effects (Example 3.5.1), the EM algorithm is useful for computing ML estimates. (In fact, it is very useful in many mixed models;

see Searle et al. 1992, Chapter 8.) Suppose we have the model Xij = µ + Ai + Uij (j = 1,...,ni, i = 1,...,s)

where Ai and Uij are independent normal random variables with mean zero and known variance. To compute the ML estimates of µ, σ2 U , and σ2 U it is typical to employ an EM algorithm using the unobservable Ai’s as the augmented data. Write out both the E-step and the M-step, and show that the EM sequence converges to the ML estimators.