Page 328 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

which is equivalent to the selection index. Thus BLUP is the selection index with the
GLS solution of b substituted for b.

ˆ
C.2 Proof that b and a ˆ from MME are the GLS
of b and BLUP of a, Respectively

In computation terms, the use of Eqn c.3 to obtain the BLUP of k′b + a is not feasible
because the inverse of V is required. Henderson (1950) formulated the MME that are
ˆ
suitable for calculating solutions for b and a, and showed later that k′b and a, where
ˆ
ˆ
b and aˆ are solutions from the MME, are the best linear unbiased estimator (BLUE)
of k′b and BLUP of a, respectively.
The usual MME for Eqn c.1 are:
⎡ X R X X R Z′⎤ ⎡ b⎤ ⎡ X R y ⎤ (c.5)
′
′
′
ˆ
−1
−1
−1
=
⎢ −1 −1 −1⎥ ⎢ ⎥ ⎢ −1 ⎥
′
′
a ˆ
+
R
⎣ Z R X Z R Z G ⎦ ⎣ ⎦ ⎣ Z′R y ⎦
ˆ
ˆ
The proof that b from the MME is the GLS of b and therefore k′b is the BLUE
of k′b was given by Henderson et al. (1959). From the second row of Eqn c.5:
ˆ
−1
−1
−1
(Z′R Z + G )aˆ = Z′R (y − Xb)
ˆ
−1
−1 −1
−1
â = (Z′R Z + G ) Z′R (y − Xb) (c.6)
From the first row of Eqn c.5:
−1
−1
X′R Xb + Z′R Zâ = X′R y
−1
Substituting the solution for aˆ into the above equation gives:
−1
−1
−1
−1
X′R Xb + X′R Z(WZ′R )(y − Xb) = X′R y
−1 −1
−1
where W = (Z′R Z + G ) :
−1
−1
−1
−1
−1
−1
X′R Xb − (X′R Z)(WZ′R )Xb = X′R y − X′R ZWZ′R y
−1
−1
−1
−1
−1
−1
X′(R − R ZWZ′R )Xb = X′(R − R ZWZ′R )y
−1
−1
X′V Xb = X′V y
−1
−1
−1
−1
with V = R – R ZWZ′R :
ˆ
−1
−1
−1
b = (X′V X) X′V y (c.7)
It can be shown that:
−1
−1
−1
V = R − R ZWZ′R −1
by pre-multiplying the right-hand side by V and obtaining an identity matrix
(Henderson et al., 1959):
−1
−1
−1
−1
−1
−1
V[R − R ZWZ′R ] = (R + ZGZ′)(R − R ZWZ′R )
−1
−1
−1
= I + ZGZ′R − ZWZ′R − ZGZ′R ZWZ′R −1
= I + ZGZ′R − Z(I + GZ′RZ)WZ′R −1
−1
−1
−1
= I + ZGZ′R − ZG(G + Z′RZ)WZ′R −1
= I + ZGZ′R − ZG(W )WZ′R −1
−1
−1
= I + ZGZ′R − ZGZ′R −1
−1
= I
312 Appendix C