Prediction Theory 1 Introduction 2 General Linear Mixed Model

Then
GZ ′ V −1 = GZ ′ (R −1 − R −1 ZTZ ′ R −1 )
= (GZ ′ R −1 − GZ ′ R −1 ZTZ ′ R −1 )
= (GT −1 − GZ ′ R −1 Z)TZ ′ R −1
= (G(Z ′ R −1 Z + G −1 ) − GZ ′ R −1 Z)TZ ′ R −1
= TZ ′ R −1 .
Similarly, the MME solution for û and substituting it into the first equation in the MME
gives
X ′ R −1 Xˆb + X ′ R −1 Z(TZ ′ R −1 (y − Xˆb)) = X ′ R −1 y.
Combine the terms in ˆb and y to give
X ′ (R −1 − R −1 ZTZ ′ R −1 )Xˆb = X ′ (R −1 − R −1 ZTZ ′ R −1 )y,
which are the same as the GLS equations,
X ′ V −1 Xˆb = X ′ V −1 y.
Goldberger (1962) published these results before Henderson (1963), but Henderson knew of
these equivalences back in 1949 through numerical examples. After he discovered Goldberger’s
paper (sometime after his retirement) Henderson insisted on citing it along with his work. Most
people in animal breeding, however, refer to Henderson as the originator of this work and its
primary proponent.
9 Variances of Predictors and Prediction Errors From MME
The covariance matrices of the predictors and prediction errors can be expressed in terms of the
generalized inverse of the coefficient matrix of the MME, C. Recall that
( ) ( ) ( )
ˆb Cxx C
=
xz X ′ R −1
û C zx C zz Z ′ R −1 y,
or as
ˆb = C ′ by,
and
û = C ′ uy.
If the coefficient matrix of the MME is full rank (or a full rank subset) (to simplify the
presentation of results), then
( ) ( ) ( )
Cxx C xz X ′ R −1 X X ′ R −1 Z
I 0
C zx C zz Z ′ R −1 X Z ′ R −1 Z + G −1 =
,
0 I
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