Page 327 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Appendix C

C.1 Outline of the Derivation of the Best Linear Unbiased
Prediction (BLUP)

Consider the following linear model:
y = Xb + Za + e (c.1)
where the expectations are:

E(y) = Xb; E(a) = E(e) = 0
and:
var(a) = As = G, var(e) = R and cov(a, e) = cov(e, a) = 0
2
a
Then, as shown in Section 3.2:
var(y) = V = ZGZ′ + R, cov(y, a) = ZG and cov(y, e) = R

The prediction problem involves both b and a. Suppose we want to predict a
linear function of b and a, say k′b + a, using a linear function of y, say L′y, and k′b
is estimable. The predictor L′y is chosen such that:
E(L′y) = E(k′b + a)
that is, it is unbiased and the prediction error variance (PEV) is minimized (Henderson,
1973). Now PEV (Henderson, 1984) is:
PEV = var(L′y − k′b + a)
= var(L′y − a)
= L′var(y)L + var(a) − L′cov(y, a) − cov(a, y)L
= L′VL + G − L′ZG − ZG′L (c.2)

Minimizing PEV subject to E(L′y) = E(k′b + a) and solving (see Henderson, 1973,
1984 for details of derivation) gives:
−1
−1
−1
−1
−1
−1
−1
L′y = k′(X′V X) X′V y − GZ′V (y − X(X′V X) X′V y)
ˆ
−1
Let b= (X′V X)XV y, the generalized least square solution for b, then the predictor
−1
can be written as:
ˆ
ˆ
L′y = k′b + GZ′V (y − Xb) (c.3)
−1
which is the BLUP of k′b + a.
Note that if k′b = 0, then:
ˆ
L′y = BLUP(a) = GZ′V (y − Xb) (c.4)
−1
© R.A. Mrode 2014. Linear Models for the Prediction of Animal Breeding Values, 311
3rd Edition (R.A. Mrode)