5.5 Consider the setting of Section 5.9 with n = 3. Let X N3(0, ) and suppose the covariance matrix satisfies ????11 =

5.5 Consider the setting of Section 5.9 with n = 3. Let X ∼ N3(0, Σ) and suppose the covariance matrix Σ satisfies ????11 = ????22 = ????33 = 1. In the notation of this section, show that the coefficient vectors ????T i = (????i1, ????i2, ????i3) are given by

????11 = 1, ????12 = 0, ????13 = 0,

????21 = −????12, ????22 = 1, ????23 = 0,

????31 = ????23????12 − ????31 1 − ????2 12

, ????32 = ????13????12 − ????23 1 − ????2 12

, ????33 = 1.

Set ej = ????T j x, j = 1, 2, 3. Show from first principles that e1, e2, and e3 are uncorrelated.

In the REML setting of Section 5.10, the second and third coefficient vectors are now required to be orthogonal to the constant vector, (????U 2 )

T???? = 0 and

(????U 3 )

T???? = 0. Show that

????U 21 = −1, ????U 22 = 1, ????U 23 = 0,

????U 31 = −1 2

(

1 + ????31 − ????32 1 − ????12 )

, ????U 32 = −1 2

(

1 − ????31 − ????32 1 − ????12 )

, ????U 33 = 1.

Note that ????U 31 + ????U 32 = −1. Further, show that eU 2 = (????U 2 )

Tx and eU 3 = (????U 3 )

Tx are uncorrelated.