Continuing Problem 5, suppose that one or more of the covariance matrices k is singular. For instance,

Continuing Problem 5, suppose that one or more of the covariance matrices Γk is singular. For instance, in modeling common household effects, the corresponding missing data Xk can be represented as Xk = σkMkWk, where Mk is a constant m×s matrix having exactly one entry 1 and the remaining entries 0 in each row, and where Wk has s independent, standard normal components. Each component of Wk corresponds to a different household; each row of Mk chooses the correct household for a given person. It follows from this description that

σ2 kΓk = Var(Xk)

= σ2 kMkMt k

The matrix Hk = MkMt k is the household indicator matrix described in the text. When Xk has the representation σkMkWk, one should replace Xk in the complete data by σkWk. With this change, show that the EM update for σ2 k is σ2 n+1,k = 1 s [tr(Υnk) + νt nkνnk], where νnk and Υnk are νk = σ2 kMt kΩ−1(y − Aµ)
Υk = σ2 kI − σ4 kMt kΩ−1Mk evaluated at the current parameter vector γn.