Exercise 4.1 To develop some facility with this notation, let Tr = [ti j], where ti j =δir − Nr n

for r = 1, . . . ,t. Find T

r Tr, T

r Ts for s = r, and JTr.

A very important application of this notation is in characterizing the vector MY.

As discussed in Section 3.1, the vector MY is the base from which all estimates of parametric functions are found. A second important application involves the projection operator Mα = M− 1 n

Jn n .

Mα is useful in testing hypotheses and is especially important in the analysis of multifactor ANOVAs. It is therefore necessary to have a characterization of Mα .