Exercise 7.7.4 Consider the linear model yi j = +i+ j+ei j, i=1, . . . ,a,

Exercise 7.7.4 Consider the linear model yi j =μ +αi+η j+ei j, i=1, . . . ,a, j =

1, . . . ,b. As in Section 1, write X = [X0,X1, . . . ,Xa,Xa+1, . . . ,Xa+b]. If we write the observations in the usual order, we can use Kronecker products to write the model matrix. Write X = [J,X∗,X∗∗], where X∗ = [X1, . . . ,Xa], and X∗∗ = [Xa+1, . . . ,Xa+b].

Using Kronecker products, X∗ = [Ia ⊗Jb], and X∗∗ = [Ja ⊗Ib]. In fact, with n = ab, J = Jn = [Ja⊗Jb]. Use Kronecker products to show that X

∗ (I−[1/n]Jn n )X∗∗ = 0. In 1 a a+1, . . . ,Za+b).

Also show that [(1/a)Ja a

⊗Ib] is the perpendicular projection operator onto C(X∗∗)

and that Mη = [(1/a)Ja a

⊗(Ib−(1/b)Jb b )].