Prediction Theory 1 Introduction 2 General Linear Mixed Model

4.4 Function to be Minimized
Because the predictor is required to be unbiased, then the mean squared error is equivalent
to the variance of prediction error. Combine the variance of prediction error with a LaGrange
Multiplier to force unbiasedness to obtain the matrix F, where
F = V ar(P E) + (L ′ X − K ′ )Φ.
Minimization of the diagonals of F is achieved by differentiating F with respect to the unknowns,
L and Φ, and equating the partial derivatives to null matrices.
∂F
∂L
= 2VL − 2ZGM + XΦ = 0
∂F
∂Φ = X′ L − K = 0
Let θ = .5Φ, then the first derivative can be written as
then solve for L as
VL = ZGM − Xθ
V −1 VL = L
= V −1 ZGM − V −1 Xθ.
Substituting the above for L into the second derivative, then we can solve for θ as
X ′ L − K = 0
X ′ (V −1 ZGM − V −1 Xθ) − K = 0
X ′ V −1 Xθ = X ′ V −1 ZGM − K
Substituting this solution for θ into the equation for L gives
θ = (X ′ V −1 X) − (X ′ V −1 ZGM − K)
L ′ = M ′ GZ ′ V −1 + K ′ (X ′ V −1 X) − X ′ V −1
−M ′ GZ ′ V −1 X(X ′ V −1 X) − X ′ V −1 .
Let
ˆb = (X ′ V −1 X) − X ′ V −1 y,
then the predictor becomes
L ′ y = K ′ˆb + M ′ GZ ′ V −1 (y − Xˆb)
which is the BLUP of K ′ b + M ′ u, and ˆb is a GLS solution for b. A special case for this predictor
would be to let K ′ = 0 and M ′ = I, then the predictand is K ′ b + M ′ u = u, and
L ′ y = û = GZ ′ V −1 (y − Xˆb).
( )
Hence the predictor of K ′ b + M ′ u is K ′ M ′ times the predictor of
(
ˆb′ û ′ ) ′
.
5
(
b ′
u ′ ) ′
which is